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We prove that for every graph $G$ with a sufficiently large complete bipartite induced minor, either $G$ has an induced minor isomorphic to a large wall, or $G$ contains a large constellation; that is, a complete bipartite induced minor model such that on one side of the bipartition, each branch set is a singleton, and on the other side, each branch set induces a path. We further refine this theorem by characterizing the unavoidable induced subgraphs of large constellations as two types of highly structured constellations. These results will be key ingredients in several forthcoming papers of this series.

We show that for any positive integer $r$, the Weighted Maximum $k$-colourable Induced Subgraph problem can be solved in polynomial time for input graphs that do not contain $(P_5+ rK_1)$ as an induced subgraph, and give an explicit algorithm demonstrating this. This answers a question of Agrawal et al. (2024).

Given a graph $G$, we say that an orientation $D$ of $G$ is a KT orientation if, for all $u, v \in V(D)$, there is at most one directed path (in any direction) between $u$ and $v$. Graphs that admit such orientations have been used by Kierstead and Trotter (1992), Carbonero, Hompe, Moore, and Spirkl (2023), Bria\'nski, Davies, and Walczak (2024), and Gir\~ao, Illingworth, Powierski, Savery, Scott, Tamitegami, and Tan (2024) to construct graphs with large chromatic number and small clique number that served as counterexamples to various conjectures. Motivated by this, we consider which graphs admit KT orientations (named after Kierstead and Trotter). In particular, we construct a graph family with small independence number (sublinear in the number of vertices) which admits a KT orientation. We show that the problem of determining whether a given graph admits a KT orientation is NP-complete, even if we restrict ourselves to planar graphs. Finally, we provide an algorithm to decide if a graph with maximum degree at most 3 admits a KT orientation, whereas, for graphs with maximum degree 4, the problem remains NP-complete.

We prove that the tree independence number of every even-hole-free graph is at most polylogarithmic in its number of vertices. More explicitly, we prove that there exists a constant c>0 such that for every integer n>1 every n-vertex even-hole-free graph has a tree decomposition where each bag has stability (independence) number at most c log^10 n. This implies that the Maximum Weight Independent Set problem, as well as several other natural algorithmic problems that are known to be NP-hard in general, can be solved in quasi-polynomial time if the input graph is even-hole-free.

A three-path-configuration is a graph consisting of three pairwise internally-disjoint paths the union of every two of which is an induced cycle of length at least four. A graph is 3PC-free if no induced subgraph of it is a three-path-configuration. We prove that 3PC-free graphs have poly-logarithmic tree-independence number. More explicitly, we show that there exists a constant $c$ such that every $n$-vertex 3PC-free graph graph has a tree decomposition in which every bag has stability number at most $c (\log n)^2$. This implies that the Maximum Weight Independent Set problem, as well as several other natural algorithmic problems, that are known to be NP-hard in general, can be solved in quasi-polynomial time if the input graph is 3PC-free.

For a property $\mathcal{P}$ of graphs, the $\mathcal{P}$-\textsc{Sandwich-Problem}, introduced by Golumbic and Shamir (1993), is the following: Given a pair of graphs $(G_1, G_2)$ on the same vertex set $V$, does there exist a graph $G$ such that $V(G)=V$, $E(G_{1})\subseteq E(G) \subseteq E(G_{2})$, and $G$ satisfies $\mathcal{P}$? A {\em hole} in a graph is an induced subgraph which is a cycle of length at least four. An odd (respectively even) hole is a hole of odd (respectively even) length. Given a class of graphs $\mathcal{C}$ and a graph $G$ we say that $G$ is {\em $\mathcal{C}$-free} if it contains no induced subgraph isomorphic to a member of $\mathcal{C}$. In this paper we prove that if $\mathcal{P}$ is the property of being odd-hole-free or the property of being even-hole-free, then the $\mathcal{P}$-\textsc{Sandwich-Problem} is NP-hard.

A hole in a graph is an induced subgraph which is a cycle of length at least four. A graph is chordal if it contains no holes. Following McKee and Scheinerman (1993), we define the chordality of a graph $G$ to be the minimum number of chordal graphs on $V(G)$ such that the intersection of their edge sets is equal to $E(G)$. In this paper we study classes of graphs of bounded chordality. In the 1970s, Buneman, Gavril, and Walter, proved independently that chordal graphs are exactly the intersection graphs of subtrees in trees. We generalize this result by proving that the graphs of chordality at most $k$ are exactly the intersection graphs of convex subgraphs of median graphs of tree-dimension $k$. A hereditary class of graphs $\mathcal{A}$ is $\chi$-bounded if there exists a function $ f\colon \mathbb{N}\rightarrow \mathbb{R}$ such that for every graph $G\in \mathcal{A}$, we have $\chi(G) \leq f(\omega(G))$. In 1960, Asplund and Gr\"unbaum proved that the class of all graphs of boxicity at most two is $\chi$-bounded. In his seminal paper "Problems from the world surrounding perfect graphs," Gy\'arf\'as (1985), motivated by the above result, asked whether the class of all graphs of chordality at most two, which we denote by $\mathcal{C}\gcap \mathcal{C}$, is $\chi$-bounded. We discuss a result of Felsner, Joret, Micek, Trotter and Wiechert (2017), concerning tree-decompositions of Burling graphs, which implies an answer to Gy\'arf\'as' question in the negative. We prove that two natural families of subclasses of $\mathcal{C}\gcap \mathcal{C}$ are polynomially $\chi$-bounded. Finally, we prove that for every $k\geq 3$ the $k$-\textsc{Chordality Problem}, which asks to decide whether a graph has chordality at most $k$, is \textbf{NP}-complete.

We prove that for every integer $t\geq 1$ there exists an integer $c_t\geq 1$ such that every $n$-vertex even-hole-free graph with no clique of size $t$ has treewidth at most $c_t\log{n}$. This resolves a conjecture of Sintiari and Trotignon, who also proved that the logarithmic bound is asymptotically best possible. It follows that several \textsf{NP}-hard problems such as \textsc{Stable Set}, \textsc{Vertex Cover}, \textsc{Dominating Set} and \textsc{Coloring} admit polynomial-time algorithms on this class of graphs. As a consequence, for every positive integer $r$, $r$-{\sc Coloring} can be solved in polynomial time on even-hole-free graphs without any assumptions on clique size. As part of the proof, we show that there is an integer $d$ such that every even-hole-free graph has a balanced separator which is contained in the (closed) neighborhood of at most $d$ vertices. This is of independent interest; for instance, it implies the existence of efficient approximation algorithms for certain \textsf{NP}-hard problems while restricted to the class of all even-hole-free graphs., Comment: arXiv admin note: text overlap with arXiv:2307.13684

The decomposition of complex networks into smaller, interconnected components is a central challenge in network theory with a wide range of potential applications. In this paper, we utilize tools from group theory and ring theory to study this problem when the network is a Cayley graph. In particular, we answer the following question: Which Cayley graphs are prime?

A clock is a graph consisting of an induced cycle $C$ and a vertex not in $C$ with at least two non-adjacent neighbours in $C$. We show that every clock-free graph of large treewidth contains a "basic obstruction" of large treewidth as an induced subgraph: a complete graph, a subdivision of a wall, or the line graph of a subdivision of a wall.

Given an integer $k>4$ and a graph $H$, we prove that, assuming P$\neq$NP, the List-$k$-Coloring Problem restricted to $H$-free graphs can be solved in polynomial time if and only if either every component of $H$ is a path on at most three vertices, or removing the isolated vertices of $H$ leaves an induced subgraph of the five-vertex path. In fact, the "if" implication holds for all $k\geq 1$.

Unlike minors, the induced subgraph obstructions to bounded treewidth come in a large variety, including, for every $t\geq 1$, the $t$-basic obstructions: the graphs $K_{t+1}$ and $K_{t,t}$, along with the subdivisions of the $t$-by-$t$ wall and their line graphs. But this list is far from complete. The simplest example of a ''non-basic'' obstruction is due to Pohoata and Davies (independently). For every $n \geq 1$, they construct certain graphs of treewidth $n$ and with no $3$-basic obstruction as an induced subgraph, which we call $n$-arrays. Let us say a graph class $\mathcal{G}$ is clean if the only obstructions to bounded treewidth in $\mathcal{G}$ are in fact the basic ones. It follows that a full description of the induced subgraph obstructions to bounded treewidth is equivalent to a characterization of all families $\mathcal{H}$ of graphs for which the class of all $\mathcal{H}$-free graphs is clean (a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to any graph in $\mathcal{H}$). This remains elusive, but there is an immediate necessary condition: if $\mathcal{H}$-free graphs are clean, then there are only finitely many integers $n\geq 1$ such that there is an $n$-array which is $\mathcal{H}$-free. The above necessary condition is not sufficient in general. However, the situation turns out to be different if $\mathcal{H}$ is finite: we prove that for every finite set $\mathcal{H}$ of graphs, the class of all $\mathcal{H}$-free graphs is clean if and only if there is no $\mathcal{H}$-free $n$-array except possibly for finitely many values of $n$.

Given an integer $c\geq 1$, we say a graph $G$ is $c$-pinched if $G$ does not contain an induced subgraph consisting of $c$ cycles, all going through a single common vertex and otherwise pairwise disjoint and with no edges between them. What can be said about the structure of $c$-pinched graphs? For instance, $1$-pinched graphs are exactly graphs of treewidth $1$. However, bounded treewidth for $c>1$ is immediately seen to be a false hope because complete graphs, complete bipartite graphs, subdivided walls and line-graphs of subdivided walls are all examples of $2$-pinched graphs with arbitrarily large treewidth. There is even a fifth obstruction for larger values of $c$, discovered by Pohoata and later independently by Davies, consisting of $3$-pinched graphs with unbounded treewidth and no large induced subgraph isomorphic to any of the first four obstructions. We fuse the above five examples into a grid-type theorem fully describing the unavoidable induced subgraphs of pinched graphs with large treewidth. More precisely, we prove that for every integer $c\geq 1$, a $c$-pinched graph $G$ has large treewidth if and only if $G$ contains one of the following as an induced subgraph: a large complete graph, a large complete bipartite graph, a subdivision of a large wall, the line-graph of a subdivision of a large wall, or a large graph from the Pohoata-Davies construction. Our main result also generalizes to an extension of pinched graphs where the lengths of excluded cycles are lower-bounded., Comment: arXiv admin note: text overlap with arXiv:2305.15615

Treewidth is a fundamental graph parameter that quantifies the complexity of graphs by measuring their "thickness" compared to a tree. In 1986, Robertson and Seymour proved that the only way a graph may have large treewidth is by containing a large hexagonal grid as a minor, or equivalently, as a (subdivided) subgraph. Which induced subgraphs prevent a graph from having small treewidth? Along with complete graphs, there are three other "basic" obstructions that should appear in the answer. But the full list remains beyond the reach of current methods, and "even holes" seem to be the crux of this matter. The reason lies in the observation that even holes are the only (non-trivial) induced subgraphs that all basic obstructions except for complete graphs have in common, and yet graphs with no even hole and no $4$-vertex complete subgraph fail to have bounded treewidth. The only known example of (even-hole, $K_4$)-free graphs of unbounded treewidth, called "layered wheels", is due to Sintiari and Trotignon. While rather complicated in global structure, layered wheels are unexpectedly sparse from a local perspective: for every $h\geq 1$, there are layered wheels of arbitrarily large treewidth in which every induced subgraph on at most $h$ vertices is chordal. The converse is also true, that every $K_4$-free chordal graph $H$ appears as an induced subgraph in every layered wheel of sufficiently large treewidth. It turns out, despite their technically involved construction, that layered wheels are in fact quite canonical: our main result shows that the local structure of layered wheels is realized in every (even-hole, $K_4$)-free graph of large treewidth. More precisely, we prove that for a graph $H$, every (even-hole, $K_4$)-free graph of large enough treewidth contains an induced subgraph isomorphic to $H$, if and only if $H$ is a $K_4$-free chordal graph.

A generalized $t$-pyramid is a graph obtained from a certain kind of tree (a subdivided star or a subdivided cubic caterpillar) and the line graph of a subdivided cubic caterpillar by identifying simplicial vertices. We prove that for every integer $t$ there exists a constant $c(t)$ such that every $n$-vertex even-hole-free graph with no clique of size $t$ and no induced subgraph isomorphic to a generalized $t$-pyramid has treewidth at most $c(t)\log{n}$. This settles a special case of a conjecture of Sintiari and Trotignon; this bound is also best possible for the class. It follows that several \textsf{NP}-hard problems such as \textsc{Stable Set}, \textsc{Vertex Cover}, \textsc{Dominating Set} and \textsc{Coloring} admit polynomial-time algorithms on this class of graphs. Results from this paper are also used in later papers of the series, in particular to solve the full version of the Sintiari-Trotignon conjecture.

We continue a line of research which studies which hereditary families of digraphs have bounded dichromatic number. For a class of digraphs $\mathcal{C}$, a hero in $\mathcal{C}$ is any digraph $H$ such that $H$-free digraphs in $\mathcal{C}$ have bounded dichromatic number. We show that if $F$ is an oriented star of degree at least five, the only heroes for the class of $F$-free digraphs are transitive tournaments. For oriented stars $F$ of degree exactly four, we show the only heroes in $F$-free digraphs are transitive tournaments, or possibly special joins of transitive tournaments. Aboulker et al. characterized the set of heroes of $\{H, K_{1} + \vec{P_{2}}\}$-free digraphs almost completely, and we show the same characterization for the class of $\{H, rK_{1} + \vec{P_{3}}\}$-free digraphs. Lastly, we show that if we forbid two "valid" orientations of brooms, then every transitive tournament is a hero for this class of digraphs.

The tree-independence number tree-$\alpha$, first defined and studied by Dallard, Milani\v{c} and \v{S}torgel, is a variant of treewidth tailored to solving the maximum independent set problem. Over a series of papers, Abrishami et al. developed the so-called central bag method to study induced obstructions to bounded treewidth. Among others, they showed that, in a certain superclass $\mathcal C$ of (even hole, diamond, pyramid)-free graphs, treewidth is bounded by a function of the clique number. In this paper, we relax the bounded clique number assumption, and show that $\mathcal C$ has bounded tree-$\alpha$. Via existing results, this yields a polynomial time algorithm for the maximum independent set problem in this class. Our result also corroborates, for this class of graphs, a conjecture of Dallard, Milani\v{c} and \v{S}torgel that in a hereditary graph class, tree-$\alpha$ is bounded if and only if the treewidth is bounded by a function of the clique number., Comment: 17 pages. arXiv admin note: text overlap with arXiv:2203.06775

The celebrated Erd\H{o}s-P\'{o}sa Theorem, in one formulation, asserts that for every $c\geq 1$, graphs with no subgraph (or equivalently, minor) isomorphic to the disjoint union of $c$ cycles have bounded treewidth. What can we say about the treewidth of graphs containing no induced subgraph isomorphic to the disjoint union of $c$ cycles? Let us call these graphs $c$-perforated. While $1$-perforated graphs have treewidth one, complete graphs and complete bipartite graphs are examples of $2$-perforated graphs with arbitrarily large treewidth. But there are sparse examples, too: Bonamy, Bonnet, D\'{e}pr\'{e}s, Esperet, Geniet, Hilaire, Thomass\'{e} and Wesolek constructed $2$-perforated graphs with arbitrarily large treewidth and no induced subgraph isomorphic to $K_3$ or $K_{3,3}$; we call these graphs occultations. Indeed, it turns out that a mild (and inevitable) adjustment of occultations provides examples of $2$-perforated graphs with arbitrarily large treewidth and arbitrarily large girth, which we refer to as full occultations. Our main result shows that the converse also holds: for every $c\geq 1$, a $c$-perforated graph has large treewidth if and only if it contains, as an induced subgraph, either a large complete graph, or a large complete bipartite graph, or a large full occultation. This distinguishes $c$-perforated graphs, among graph classes purely defined by forbidden induced subgraphs, as the first to admit a grid-type theorem incorporating obstructions other than subdivided walls and their line graphs. More generally, for all $c,o\geq 1$, we establish a full characterization of induced subgraph obstructions to bounded treewidth in graphs containing no induced subgraph isomorphic to the disjoint union of $c$ cycles, each of length at least $o+2$.

We prove that every $P_5$-free graph of bounded clique number contains a small hitting set of all its maximum stable sets. More generally, let us say a class $\mathcal{C}$ of graphs is $\eta$-bounded if there exists a function $h:\mathbb{N}\rightarrow \mathbb{N}$ such that $\eta(G)\leq h(\omega(G))$ for every graph $G\in \mathcal{C}$, where $\eta(G)$ denotes smallest cardinality of a hitting set of all maximum stable sets in $G$, and $\omega(G)$ is the clique number of $G$. Also, $\mathcal{C}$ is said to be polynomially $\eta$-bounded if in addition $h$ can be chosen to be a polynomial. We introduce $\eta$-boundedness inspired by a question of Alon and motivated by a number of meaningful similarities to $\chi$-boundedness. In particular, we propose an analogue of the Gy\'{a}rf\'{a}s-Sumner conjecture, that the class of all $H$-free graphs is $\eta$-bounded if (and only if) $H$ is a forest. Like $\chi$-boundedness, the case where $H$ is a star is easy to verify, and we prove two non-trivial extensions of this: $H$-free graphs are $\eta$-bounded if (1) $H$ has a vertex incident with all edges of $H$, or (2) $H$ can be obtained from a star by subdividing at most one edge, exactly once. Unlike $\chi$-boundedness, the case where $H$ is a path is surprisingly hard. Our main result mentioned at the beginning shows that $P_5$-free graphs are $\eta$-bounded. The proof is rather involved compared to the classical ``Gy\'{a}rf\'{a}s path'' argument which establishes, for all $t$, the $\chi$-boundedness of $P_t$-free graphs. It remains open whether $P_t$-free graphs are $\eta$-bounded for $t\geq 6$. It also remains open whether $P_5$-free graphs are polynomially $\eta$-bounded, which, if true, would imply the Erd\H{o}s-Hajnal conjecture for $P_5$-free graphs. But we prove that $H$-free graphs are polynomially $\eta$-bounded if $H$ is a proper induced subgraph of $P_5$., Comment: [v2] Accepted manuscript; see DOI for journal version [v3] Fixed mistake in (16)

Schur functions are a basis of the symmetric function ring that represent Schubert cohomology classes for Grassmannians. Replacing the cohomology ring with $K$-theory yields a rich combinatorial theory of inhomogeneous deformations, where Schur functions are replaced by their $K$-analogues, the basis of symmetric Grothendieck functions. We introduce and initiate a theory of the Kromatic symmetric function $\overline{X}_G$, a $K$-theoretic analogue of the chromatic symmetric function $X_G$ of a graph $G$. The Kromatic symmetric function is a generating series for graph colorings in which vertices may receive any nonempty set of distinct colors such that neighboring color sets are disjoint. Our main result lifts a theorem of Gasharov (1996) to this setting, showing that when $G$ is a claw-free incomparability graph, $\overline{X}_G$ is a positive sum of symmetric Grothendieck functions. This result suggests a topological interpretation of Gasharov's theorem. We then show that the Kromatic symmetric functions of path graphs are not positive in any of several $K$-analogues of the $e$-basis of symmetric functions, demonstrating that the Stanley-Stembridge conjecture (1993) does not have such a lift to $K$-theory and so is unlikely to be amenable to a topological perspective. We also define a vertex-weighted extension of $\overline{X}_G$ and show that it admits a deletion--contraction relation. Finally, we give a $K$-analogue for $\overline{X}_G$ of the classic monomial-basis expansion of $X_G$., Comment: Updated theorem statement and proof for Grothendieck-positivity of Kromatic symmetric functions; an exact combinatorial interpretation is now provided

Given a graph $H$, we prove that every (theta, prism)-free graph of sufficiently large treewidth contains either a large clique or an induced subgraph isomorphic to $H$, if and only if $H$ is a forest.

We say a class $\mathcal{C}$ of graphs is clean if for every positive integer $t$ there exists a positive integer $w(t)$ such that every graph in $\mathcal{C}$ with treewidth more than $w(t)$ contains an induced subgraph isomorphic to one of the following: the complete graph $K_t$, the complete bipartite graph $K_{t,t}$, a subdivision of the $(t\times t)$-wall or the line graph of a subdivision of the $(t \times t)$-wall. In this paper, we adapt a method due to Lozin and Razgon (building on earlier ideas of Wei{\ss}auer) to prove that the class of all $H$-free graphs (that is, graphs with no induced subgraph isomorphic to a fixed graph $H$) is clean if and only if $H$ is a forest whose components are subdivided stars. Their method is readily applied to yield the above characterization. However, our main result is much stronger: for every forest $H$ as above, we show that forbidding certain connected graphs containing $H$ as an induced subgraph (rather than $H$ itself) is enough to obtain a clean class of graphs. Along the proof of the latter strengthening, we build on a result of Davies and produce, for every positive integer $\eta$, a complete description of unavoidable connected induced subgraphs of a connected graph $G$ containing $\eta$ vertices from a suitably large given set of vertices in $G$. This is of independent interest, and will be used in subsequent papers in this series., Comment: Accepted manuscript; see DOI for journal version

In this paper, we extend the chromatic symmetric function $X$ to a chromatic $k$-multisymmetric function $X_k$, defined for graphs equipped with a partition of their vertex set into $k$ parts. We demonstrate that this new function retains the basic properties and basis expansions of $X$, and we give a method for systematically deriving new linear relationships for $X$ from previous ones by passing them through $X_k$. In particular, we show how to take advantage of homogeneous sets of $G$ (those $S \subseteq V(G)$ such that each vertex of $V(G) \backslash S$ is either adjacent to all of $S$ or is nonadjacent to all of $S$) to relate the chromatic symmetric function of $G$ to those of simpler graphs. Furthermore, we show how extending this idea to homogeneous pairs $S_1 \sqcup S_2 \subseteq V(G)$ generalizes the process used by Guay-Paquet to reduce the Stanley-Stembridge conjecture to unit interval graphs.

This paper continues a series of papers investigating the following question: which hereditary graph classes have bounded treewidth? We call a graph $t$-clean if it does not contain as an induced subgraph the complete graph $K_t$, the complete bipartite graph $K_{t, t}$, subdivisions of a $(t \times t)$-wall, and line graphs of subdivisions of a $(t \times t)$-wall. It is known that graphs with bounded treewidth must be $t$-clean for some $t$; however, it is not true that every $t$-clean graph has bounded treewidth. In this paper, we show that three types of cutsets, namely clique cutsets, 2-cutsets, and 1-joins, interact well with treewidth and with each other, so graphs that are decomposable by these cutsets into basic classes of bounded treewidth have bounded treewidth. We apply this result to two hereditary graph classes, the class of ($ISK_4$, wheel)-free graphs and the class of graphs with no cycle with a unique chord. These classes were previously studied and decomposition theorems were obtained for both classes. Our main results are that $t$-clean ($ISK_4$, wheel)-free graphs have bounded treewidth and that $t$-clean graphs with no cycle with a unique chord have bounded treewidth., Comment: Accepted manuscript; see DOI for journal version

For a digraph $G$ and $v \in V(G)$, let $\delta^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-H\"{a}ggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $\delta^+(v) \ge k$ for all $v \in V(G)$, then $G$ contains a directed cycle of length at most $\lceil n/k \rceil$. Aharoni proposed a generalization of this conjecture, that a simple edge-colored graph on $n$ vertices with $n$ color classes, each of size at least $k$, has a rainbow cycle of length at most $\lceil n/k \rceil$. Let us call $(\alpha, \beta)$ \emph{triangular} if every simple edge-colored graph on $n$ vertices with at least $\alpha n$ color classes, each with at least $\beta n$ edges, has a rainbow triangle. Aharoni, Holzman, and DeVos showed the following: $(9/8,1/3)$ is triangular; $(1,2/5)$ is triangular. In this paper, we improve those bounds, showing the following: $(1.1077,1/3)$ is triangular; $(1,0.3988)$ is triangular. Our methods give results for infinitely many pairs $(\alpha, \beta)$, including $\beta < 1/3$; we show that $(1.3481,1/4)$ is triangular., Comment: Accepted manuscript; see DOI for journal version

The List-3-Coloring Problem is to decide, given a graph $G$ and a list $L(v)\subseteq \{1,2,3\}$ of colors assigned to each vertex $v$ of $G$, whether $G$ admits a proper coloring $\phi$ with $\phi(v)\in L(v)$ for every vertex $v$ of $G$, and the $3$-Coloring Problem is the List-$3$-Coloring Problem on instances with $L(v)=\{1,2,3\}$ for every vertex $v$ of $G$. The List-$3$-Coloring Problem is a classical NP-complete problem, and it is well-known that while restricted to $H$-free graphs (meaning graphs with no induced subgraph isomorphic to a fixed graph $H$), it remains NP-complete unless $H$ is isomorphic to an induced subgraph of a path. However, the current state of art is far from proving this to be sufficient for a polynomial time algorithm; in fact, the complexity of the $3$-Coloring Problem on $P_8$-free graphs (where $P_8$ denotes the eight-vertex path) is unknown. Here we consider a variant of the List-$3$-Coloring Problem called the Ordered Graph List-$3$-Coloring Problem, where the input is an ordered graph, that is, a graph along with a linear order on its vertex set. For ordered graphs $G$ and $H$, we say $G$ is $H$-free if $H$ is not isomorphic to an induced subgraph of $G$ with the isomorphism preserving the linear order. We prove, assuming $H$ to be an ordered graph, a nearly complete dichotomy for the Ordered Graph List-$3$-Coloring Problem restricted to $H$-free ordered graphs. In particular, we show that the problem can be solved in polynomial time if $H$ has at most one edge, and remains NP-complete if $H$ has at least three edges. Moreover, in the case where $H$ has exactly two edges, we give a complete dichotomy when the two edges of $H$ share an end, and prove several NP-completeness results when the two edges of $H$ do not share an end, narrowing the open cases down to three very special types of two-edge ordered graphs., Comment: Accepted manuscript; see DOI for journal version

What are the unavoidable induced subgraphs of graphs with large treewidth? It is well-known that the answer must include a complete graph, a complete bipartite graph, all subdivisions of a wall and line graphs of all subdivisions of a wall (we refer to these graphs as the "basic treewidth obstructions"). So it is natural to ask whether graphs excluding the basic treewidth obstructions as induced subgraphs have bounded treewidth. Sintiari and Trotignon answered this question in the negative. Their counterexamples, the so-called "layered wheels," contain wheels, where a wheel consists of a hole (i.e., an induced cycle of length at least four) along with a vertex with at least three neighbors in the hole. This leads one to ask whether graphs excluding wheels and the basic treewidth obstructions as induced subgraphs have bounded treewidth. This also turns out to be false due to Davies' recent example of graphs with large treewidth, no wheels and and no basic treewidth obstructions as induced subgraphs. However, in Davies' example there exist holes and vertices (outside of the hole) with two neighbors in them. Here we prove that a hole with a vertex with at least two neighbors in it is inevitable in graphs with large treewidth and no basic obstruction. Our main result is that graphs in which every vertex has at most one neighbor in every hole (that does not contain it) and with the basic treewidth obstructions excluded as induced subgraphs have bounded treewidth.

A theorem of Mader shows that every graph with average degree at least eight has a $K_6$ minor, and this is false if we replace eight by any smaller constant. Replacing average degree by minimum degree seems to make little difference: we do not know whether all graphs with minimum degree at least seven have $K_6$ minors, but minimum degree six is certainly not enough. For every $c>0$ there are arbitrarily large graphs with average degree at least $8-c$ and minimum degree at least six, with no $K_6$ minor. But what if we restrict ourselves to bipartite graphs? The first statement remains true: for every $c>0$ there are arbitrarily large bipartite graphs with average degree at least $8-c$ and no $K_6$ minor. But surprisingly, going to minimum degree now makes a significant difference. We will show that every bipartite graph with minimum degree at least six has a $K_6$ minor. Indeed, it is enough that every vertex in the larger part of the bipartition has degree at least six.

Given a graph $G$ and a graph property $P$ we say that $G$ is minimal with respect to $P$ if no proper induced subgraph of $G$ has the property $P$. An HC-obstruction is a minimal 2-connected non-Hamiltonian graph. Given a graph $H$, a graph $G$ is $H$-free if $G$ has no induced subgraph isomorphic to $H$. The main motivation for this paper originates from a theorem of Duffus, Gould, and Jacobson (1981), which characterizes all the minimal connected graphs with no Hamiltonian path. In 1998, Brousek characterized all the claw-free HC-obstructions. On a similar note, Chiba and Furuya (2021), characterized all (not only the minimal) 2-connected non-Hamiltonian $\{K_{1,3}, N_{3,1,1}\}$-free graphs. Recently, Cheriyan, Hajebi, and two of us (2022), characterized all triangle-free HC-obstructions and all the HC-obstructions which are split graphs. A wheel is a graph obtained from a cycle by adding a new vertex with at least three neighbors in the cycle. In this paper we characterize all the HC-obstructions which are wheel-free graphs., Comment: Accepted manuscript; see DOI for journal version

For $t \ge 2$, let us call a digraph $D$ \emph{t-chordal} if all induced directed cycles in $D$ have length equal to $t$. In a previous paper, we asked for which $t$ it is true that $t$-chordal graphs with bounded clique number have bounded dichromatic number. Recently, Aboulker, Bousquet, and de Verclos answered this in the negative for $t=3$, that is, they gave a construction of $3$-chordal digraphs with clique number at most $3$ and arbitrarily large dichromatic number. In this paper, we extend their result, giving for each $t \ge 3$ a construction of digraphs with clique number at most $3$ and arbitrarily large dichromatic number, thus answering our question in the negative. On the other hand, we show that a more restricted class, digraphs with no induced directed cycle of length less than $t$, and no induced directed $t$-vertex path, have bounded dichromatic number if their clique number is bounded. We also show the following complexity result: for fixed $t \ge 2$, the problem of determining whether a digraph is $t$-chordal is coNP-complete., Comment: Accepted manuscript; see DOI for journal version. One of the proofs was previously in arxiv:2201.08204, but has been moved here

A hole in a graph $G$ is an induced cycle of length at least four, and an even hole is a hole of even length. The diamond is the graph obtained from the complete graph $K_4$ by removing an edge. A pyramid is a graph consisting of a triangle called the base, a vertex called the apex, and three internally disjoint paths starting at the apex and disjoint otherwise, each joining the apex to a vertex of the base. For a family $\mathcal{H}$ of graphs, we say a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. Cameron, da Silva, Huang, and Vu\v{s}kovi\'c proved that (even hole, triangle)-free graphs have treewidth at most five, which motivates studying the treewidth of even-hole-free graphs of larger clique number. Sintiari and Trotignon provided a construction of (even hole, pyramid, $K_4$)-free graphs of arbitrarily large treewidth. Here, we show that for every $t$, (even hole, pyramid, diamond, $K_t$)-free graphs have bounded treewidth. The graphs constructed by Sintiari and Trotignon contain diamonds, so our result is sharp in the sense that it is false if we do not exclude diamonds. Our main result is in fact more general, that treewidth is bounded in graphs excluding certain wheels and three-path-configurations, diamonds, and a fixed complete graph. The proof uses "non-crossing decompositions" methods similar to those in previous papers in this series. In previous papers, however, bounded degree was a necessary condition to prove bounded treewidth. The result of this paper is the first to use the method of "non-crossing decompositions" to prove bounded treewidth in a graph class of unbounded maximum degree.

A pure pair in a tournament $G$ is an ordered pair $(A,B)$ of disjoint subsets of $V(G)$ such that every vertex in $B$ is adjacent from every vertex in $A$. Which tournaments $H$ have the property that if $G$ is a tournament not containing $H$ as a subtournament, and $|G|>1$, there is a pure pair $(A,B)$ in $G$ with $|A|,|B|\ge c|G|$, where $c>0$ is a constant independent of $G$? Let us say that such a tournament $H$ has the strong EH-property. As far as we know, it might be that a tournament $H$ has this property if and only if its vertex set has a linear ordering in which its backedges form a forest. Certainly this condition is necessary, but we are far from proving sufficiency. We make a small step in this direction, showing that if a tournament can be ordered with at most three backedges then it has the strong EH-property (except for one case, that we could not decide). In particular, every tournament with at most six vertices has the property, except for three that we could not decide. We also give a seven-vertex tournament that does not have the strong EH-property. This is related to the Erdos-Hajnal conjecture, which in one form says that for every tournament $H$ there exists $\tau>0$ such that every tournament $G$ not containing $H$ as a subtournament has a transitive subtournament of cardinality at least $|G|^\tau$. Let us say that a tournament $H$ satisfying this has the EH-property. It is known that every tournament with the strong EH-property also has the EH-property; so our result extends work by Berger, Choromanski and Chudnovsky, who proved that every tournament with at most six vertices has the EH-property, except for one that they did not decide., Comment: Accepted manuscript; see DOI for journal version

A class of graphs is $\chi$-bounded if there is a function $f$ such that every graph $G$ in the class has chromatic number at most $f(\omega(G))$, where $\omega(G)$ is the clique number of $G$; the class is polynomially $\chi$-bounded if $f$ can be taken to be a polynomial. The Gy\'arf\'as-Sumner conjecture asserts that, for every forest $H$, the class of $H$-free graphs (graphs with no induced copy of $H$) is $\chi$-bounded. Let us say a forest $H$ is good if it satisfies the stronger property that the class of $H$-free graphs is polynomially $\chi$-bounded. Very few forests are known to be good: for example, it is open for the five-vertex path. Indeed, it is not even known that if every component of a forest $H$ is good then $H$ is good, and in particular, it was not known that the disjoint union of two four-vertex paths is good. Here we show the latter, and more generally, that if $H$ is good then so is the disjoint union of $H$ and a four-vertex path. We also prove a more general result: if every component of $H_1$ is good, and $H_2$ is any path (or broom) then the class of graphs that are both $H_1$-free and $H_2$-free is polynomially $\chi$-bounded., Comment: Accepted manuscript; see DOI for journal version

A hole in a graph $G$ is an induced cycle of length at least four, and a $k$-multihole in $G$ is a set of pairwise disjoint and nonadjacent holes. It is well known that if $G$ does not contain any holes then its chromatic number is equal to its clique number. In this paper we show that, for any $k$, if $G$ does not contain a $k$-multihole, then its chromatic number is at most a polynomial function of its clique number. We show that the same result holds if we ask for all the holes to be odd or of length four; and if we ask for the holes to be longer than any fixed constant or of length four. This is part of a broader study of graph classes that are polynomially $\chi$-bounded.

We prove that for every $n$, there is a graph $G$ with $\chi(G) \geq n$ and $\omega(G) \leq 3$ such that every induced subgraph $H$ of $G$ with $\omega(H) \leq 2$ satisfies $\chi(H) \leq 4$. This disproves a well-known conjecture. Our construction is a digraph with bounded clique number, large dichromatic number, and no induced directed cycles of odd length at least 5., Comment: Accepted manuscript, see DOI for journal version

A pure pair of size $t$ in a graph $G$ is a pair $A,B$ of disjoint sets of $t$ vertices such that $A$ is either complete or anticomplete to $B$. It is known that, for every forest $H$, every graph on $n\ge2$ vertices that does not contain $H$ or its complement as an induced subgraph has a pure pair of size $\Omega(n)$; furthermore, this only holds when $H$ or its complement is a forest. In this paper, we look at pure pairs of size $n^{1-c}$, where $0c$.

Motivated by the analogous questions in graphs, we study the complexity of coloring and stable set problems in hypergraphs with forbidden substructures and bounded edge size. Letting $\nu(G)$ denote the maximum size of a matching in $H$, we obtain complete dichotomies for the complexity of the following problems parametrized by fixed $r, k, s \in \mathbb{N}$: $r$-Coloring in hypergraphs $G$ with edge size at most $k$ and $\nu(G) \leq s$; $r$-Precoloring Extension in $k$-uniform hypergraphs $G$ with $\nu(G) \leq s$; $r$-Precoloring Extension in hypergraphs $G$ with edge size at most $k$ and $\nu(G) \leq s$; Maximum Stable Set in $k$-uniform hypergraphs $G$ with $\nu(G) \leq s$; Maximum Weight Stable Set in $k$-uniform hypergraphs with $\nu(G) \leq s$; as well as partial results for $r$-Coloring in $k$-uniform hypergraphs $\nu(G) \leq s$. We then turn our attention to $2$-Coloring in 3-uniform hypergraphs with forbidden induced subhypergraphs, and give a polynomial-time algorithm when restricting the input to hypergraphs excluding a fixed one-edge hypergraph. Finally, we consider linear 3-uniform hypergraphs (in which every two edges share at most one vertex), and show that excluding an induced matching in $G$ implies that $\nu(G)$ is bounded by a constant; and that $3$-coloring linear $3$-uniform hypergraphs $G$ with $\nu(G) \leq 532$ is NP-hard., Comment: Accepted manuscript; see DOI for journal version

Fix k>0, and let G be a graph, with vertex set partitioned into k subsets (`blocks') of approximately equal size. An induced subgraph of G is transversal (with respect to this partition) if it has exactly one vertex in each block (and therefore it has exactly k vertices). A pure pair in G is a pair X,Y of disjoint subsets of V(G) such that either all edges between X,Y are present or none are; and in the present context we are interested in pure pairs (X,Y) where each of X,Y is a subset of one of the blocks, and not the same block. This paper collects several results and open questions concerning how large a pure pair must be present if various types of transversal subgraphs are excluded., Comment: Accepted manuscript; see DOI for journal version

A graph G is H-free if it has no induced subgraph isomorphic to H. We prove that a $P_5$-free graph with clique number $\omega\ge 3$ has chromatic number at most $\omega^{\log_2(\omega)}$. The best previous result was an exponential upper bound $(5/27)3^{\omega}$, due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdos-Hajnal conjecture holds for $P_5$, which is the smallest open case. Thus there is great interest in whether there is a polynomial bound for $P_5$-free graphs, and our result is an attempt to approach that.

A theta is a graph consisting of two non-adjacent vertices and three internally disjoint paths between them, each of length at least two. For a family $\mathcal{H}$ of graphs, we say a graph $G$ is $\mathcal{H}$-free if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. We prove a conjecture of Sintiari and Trotignon, that there exists an absolute constant $c$ for which every (theta, triangle)-free graph $G$ has treewidth at most $c\log (|V(G)|)$. A construction by Sintiari and Trotignon shows that this bound is asymptotically best possible, and (theta, triangle)-free graphs comprise the first known hereditary class of graphs with arbitrarily large yet logarithmic treewidth. Our main result is in fact a generalization of the above conjecture, that treewidth is at most logarithmic in $|V(G)|$ for every graph $G$ excluding the so-called three-path-configurations as well as a fixed complete graph. It follows that several NP-hard problems such as Stable Set, Vertex Cover, Dominating Set and Coloring admit polynomial time algorithms in graphs excluding the three-path-configurations and a fixed complete graph.

In 1981, Duffus, Gould, and Jacobson showed that every connected graph either has a Hamiltonian path, or contains a claw ($K_{1,3}$) or a net (a fixed six-vertex graph) as an induced subgraph. This implies that subject to being connected, these two are the only minimal (under taking induced subgraphs) graphs with no Hamiltonian path. Brousek (1998) characterized the minimal graphs that are $2$-connected, non-Hamiltonian and do not contain the claw as an induced subgraph. We characterize the minimal graphs that are $2$-connected and non-Hamiltonian for two classes of graphs: (1) split graphs, (2) triangle-free graphs. We remark that testing for Hamiltonicity is NP-hard in both of these classes., Comment: Accepted manuscript, see DOI for journal version

A double star is a tree with two internal vertices. It is known that the Gy\'arf\'as-Sumner conjecture holds for double stars, that is, for every double star $H$, there is a function $f$ such that if $G$ does not contain $H$ as an induced subgraph then $\chi(G)\le f(\omega(G))$ (where $\chi, \omega$ are the chromatic number and the clique number of $G$). Here we prove that $f$ can be chosen to be a polynomial.

Plethysm is a fundamental operation in symmetric function theory, derived directly from its connection with representation theory. However, it does not admit a simple combinatorial interpretation, and finding coefficients of Schur function plethysms is a major open question. In this paper, we introduce a graph-theoretic interpretation for any plethysm based on the chromatic symmetric function. We use this interpretation to give simple proofs of new and previously known plethystic identities, as well as chromatic symmetric function identities., Comment: Final version of manuscript; published with Electronic Journal of Combinatorics

This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that for all $k$ and $\Delta$, every graph with maximum degree at most $\Delta$ and sufficiently large treewidth contains either a subdivision of the $(k\times k)$-wall or the line graph of a subdivision of the $(k\times k)$-wall as an induced subgraph. We prove two theorems supporting this conjecture, as follows. 1. For $t\geq 2$, a $t$-theta is a graph consisting of two nonadjacent vertices and three internally disjoint paths between them, each of length at least $t$. A $t$-pyramid is a graph consisting of a vertex $v$, a triangle $B$ disjoint from $v$ and three paths starting at $v$ and disjoint otherwise, each joining $v$ to a vertex of $B$, and each of length at least $t$. We prove that for all $k,t$ and $\Delta$, every graph with maximum degree at most $\Delta$ and sufficiently large treewidth contains either a $t$-theta, or a $t$-pyramid, or the line graph of a subdivision of the $(k\times k)$-wall as an induced subgraph. This affirmatively answers a question of Pilipczuk et al. asking whether every graph of bounded maximum degree and sufficiently large treewidth contains either a theta or a triangle as an induced subgraph (where a theta means a $t$-theta for some $t\geq 2$). 2. A subcubic subdivided caterpillar is a tree of maximum degree at most three whose all vertices of degree three lie on a path. We prove that for every $\Delta$ and subcubic subdivided caterpillar $T$, every graph with maximum degree at most $\Delta$ and sufficiently large treewidth contains either a subdivision of $T$ or the line graph of a subdivision of $T$ as an induced subgraph., Comment: Accepted manuscript; see DOI for journal version

The Gyarfas-Sumner conjecture says that for every forest $H$, there is a function $f$ such that if $G$ is $H$-free then $\chi(G)\le f(\omega(G))$ (where $\chi, \omega$ are the chromatic number and the clique number of $G$). Louis Esperet conjectured that, whenever such a statement holds, $f$ can be chosen to be a polynomial. The Gyarfas-Sumner conjecture is only known to be true for a modest set of forests $H$, and Esperet's conjecture is known to be true for almost no forests. For instance, it is not known when $H$ is a five-vertex path. Here we prove Esperet's conjecture when each component of $H$ is a star.

What can be said about the structure of graphs that do not contain an induced copy of some graph H? Rodl showed in the 1980s that every H-free graph has large parts that are very dense or very sparse. More precisely, let us say that a graph F on n vertices is c-restricted if either F or its complement has maximum degree at most cn. Rodl proved that for every graph H, and every c>0, every H-free graph G has a linear-sized set of vertices inducing a c-restricted graph. We strengthen Rodl's result as follows: for every graph H, and all c>0, every H-free graph can be partitioned into a bounded number of subsets inducing c-restricted graphs.

A pure pair in a graph $G$ is a pair $A,B$ of disjoint subsets of $V(G)$ such that $A$ is complete or anticomplete to $B$. Jacob Fox showed that for all $\epsilon>0$, there is a comparability graph $G$ with $n$ vertices, where $n$ is large, in which there is no pure pair $A,B$ with $|A|,|B|\ge \epsilon n$. He also proved that for all $c>0$ there exists $\epsilon>0$ such that for every comparability graph $G$ with $n>1$ vertices, there is a pure pair $A,B$ with $|A|,|B|\ge \epsilon n^{1-c}$; and conjectured that the same holds for every perfect graph $G$. We prove this conjecture and strengthen it in several ways. In particular, we show that for all $c>0$, and all $\ell_1, \ell_2\ge 4c^{-1}+9$, there exists $\epsilon>0$ such that, if $G$ is a graph with $n>1$ vertices and no hole of length exactly $\ell_1$ and no antihole of length exactly $\ell_2$, then there is a pure pair $A,B$ in $G$ with $|A|\ge \epsilon n$ and $|B|\ge \epsilon n^{1-c}$. This is further strengthened, replacing excluding a hole by excluding some long subdivision of a general graph.

For a digraph $G$ and $v \in V(G)$, let $\delta^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-H\"{a}ggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $\delta^+(v) \ge k$ for all $v \in V(G)$, then $G$ contains a directed cycle of length at most $\lceil n/k \rceil$. Aharoni proposed a generalization of this conjecture, that a simple edge-colored graph on $n$ vertices with $n$ color classes, each of size $k$, has a rainbow cycle of length at most $\lceil n/k \rceil$. With Pelik\'anov\'a and Pokorn\'a, we showed that this conjecture is true if each color class has size ${\Omega}(k\log k)$. In this paper, we present a proof of the conjecture if each color class has size ${\Omega}(k)$, which improved the previous result and is only a constant factor away from Aharoni's conjecture. We also consider what happens when the condition on the number of colors is relaxed., Comment: Updating paper with added corrigendum section fixing error in earlier version

For a positive integer $r$ and graphs $G$ and $H$, we denote by $G+H$ the disjoint union of $G$ and $H$, and by $rH$ the union of $r$ mutually disjoint copies of $H$. Also, we say $G$ is $H$-free if $H$ is not isomorphic to an induced subgraph of $G$. We use $P_t$ to denote the path on $t$ vertices. For a fixed positive integer $k$, the List-$k$-Coloring Problem is to decide, given a graph $G$ and a list $L(v)\subseteq \{1,\ldots,k\}$ of colors assigned to each vertex $v$ of $G$, whether $G$ admits a proper coloring $\phi$ with $\phi(v)\in L(v)$ for every vertex $v$ of $G$, and the $k$-Coloring Problem is the List-$k$-Coloring Problem restricted to instances with $L(v)=\{1,\ldots, k\}$ for every vertex $v$ of $G$. We prove that for every positive integer $r$, the List-$5$-Coloring Problem restricted to $rP_3$-free graphs can be solved in polynomial time. Together with known results, this gives a complete dichotomy for the complexity of the List-$5$-Coloring Problem restricted to $H$-free graphs: For every graph $H$, assuming P$\neq$NP, the List-$5$-Coloring Problem restricted to $H$-free graphs can be solved in polynomial time if and only if $H$ is an induced subgraph of either $rP_3$ or $P_5+rP_1$ for some positive integer $r$. As a hardness counterpart, we also show that the $k$-Coloring Problem restricted to $rP_4$-free graphs is NP-complete for all $k\geq 5$ and $r\geq 2$., Comment: Accepted manuscript, see DOI for journal version