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The problem \textsc{Token Jumping} asks whether, given a graph $G$ and two independent sets of \emph{tokens} $I$ and $J$ of $G$, we can transform $I$ into $J$ by changing the position of a single token in each step and having an independent set of tokens throughout. We show that there is a polynomial-time algorithm that, given an instance of \textsc{Token Jumping}, computes an equivalent instance of size $O(g^2 + gk + k^2)$, where $g$ is the genus of the input graph and $k$ is the size of the independent sets., Comment: 11 pages, 1 figure

We study the problem of finding homomorphisms into odd cycles from planar graphs with high odd-girth. The Jaeger-Zhang conjecture states that every planar graph of odd-girth at least $4k+1$ admits a homomorphism to the odd cycle $C_{2k+1}$. The $k=1$ case is the well-known Gr\"otzsch's $3$-coloring theorem. For general $k$, in 2013 Lov\'asz, Thomassen, Wu, and Zhang showed that it suffices to have odd-girth at least $6k+1$. Improvements are known for $C_5$ and $C_7$ in [Combinatorica 2017, SIDMA 2020, Combinatorica 2022]. For $C_9$ we improve this hypothesis by showing that it suffices to have odd-girth 23. Our main tool is a variation on the potential method applied to modular orientations. This allows more flexibility when seeking reducible configurations. The same techniques also prove some results on circular coloring of signed planar graphs., Comment: 24 pages, 4 figures

Let $G$ be a planar graph and $I_s$ and $I_t$ be two independent sets in $G$, each of size $k$. We begin with a "token" on each vertex of $I_s$ and seek to move all tokens to $I_t$, by repeated "token jumping", removing a single token from one vertex and placing it on another vertex. We require that each intermediate arrangement of tokens again specifies an independent set of size $k$. Given $G$, $I_s$, and $I_t$, we ask whether there exists a sequence of token jumps that transforms $I_s$ to $I_t$. When $k$ is part of the input, this problem is known to be PSPACE-complete. However, it was shown by Ito, Kami\'nski, and Ono to be fixed-parameter tractable. That is, when $k$ is fixed, the problem can be solved in time polynomial in the order of $G$. Here we strengthen the upper bound on the running time in terms of $k$ by showing that the problem has a kernel of size linear in $k$. More precisely, we transform an arbitrary input problem on a planar graph into an equivalent problem on a (planar) graph with order $O(k)$., Comment: There is an error in the proof of Claim 7. This general approach can be salvaged to give a kernel that is quadratic in k (rather than linear). This has been done, with two coauthors, in the more general context of graphs on arbitrary surfaces in arXiv:2408.04743 [cs.DS]

For a graph $G$ and a list assignment $L$ with $|L(v)|=k$ for all $v$, an $L$-packing consists of $L$-colorings $\varphi_1,\cdots,\varphi_k$ such that $\varphi_i(v)\ne\varphi_j(v)$ for all $v$ and all distinct $i,j\in\{1,\ldots,k\}$. Let $\chi^{\star}_{\ell}(G)$ denote the smallest $k$ such that $G$ has an $L$-packing for every $L$ with $|L(v)|=k$ for all $v$. Let $\mathcal{P}_k$ denote the set of all planar graphs with girth at least $k$. We show that (i) $\chi^{\star}_{\ell}(G)\le 8$ for all $G\in \mathcal{P}_3$ and (ii) $\chi^{\star}_{\ell}(G)\le 5$ for all $G\in \mathcal{P}_4$ and (iii) $\chi^{\star}_{\ell}(G)\le 4$ for all $G\in \mathcal{P}_5$. Part (i) makes progress on a problem of Cambie, Cames van Batenburg, Davies, and Kang. We also construct outerplanar graphs $G$ such that $\chi^{\star}_{\ell}(G)=4$, which matches the known upper bound $\chi^{\star}_{\ell}(G)\le 4$ for all outerplanar graphs. Finally, we consider the analogue of $\chi^{\star}_{\ell}$ for correspondence coloring, $\chi^{\star}_c$. In fact, all bounds stated above for $\chi^{\star}_{\ell}$ also hold for $\chi^{\star}_c$., Comment: 21 pages, 16 figures

Wegner conjectured that if $G$ is a planar graph with maximum degree $\Delta\ge 8$, then $\chi(G^2)\le \left\lfloor \frac32\Delta\right\rfloor +1$. This problem has received much attention, but remains open for all $\Delta\ge 8$. Here we prove an analogous bound on $\omega(G^2)$: If $G$ is a plane graph with $\Delta(G)\ge 36$, then $\omega(G^2)\le \lfloor\frac32\Delta(G)\rfloor+1$. In fact, this is a corollary of the following lemma, which is our main result. If $G$ is a plane graph with $\Delta(G)\ge 19$ and $S$ is a maximal clique in $G^2$ with $|S|\ge \Delta(G)+20$, then there exist $x,y,z\in V(G)$ such that $S=\{w:|N[w]\cap\{x,y,z\}|\ge 2\}$., Comment: 7 pages, 5 figures, 2nd version incorporates minor reviewer feedback, to appear in European Journal of Combinatorics

Hocquard, Kim, and Pierron constructed, for every even integer $D\ge 2$, a 2-degenerate graph $G_D$ with maximum degree $D$ such that $\omega(G_D^2)=\frac52D$. We prove for (a) all 2-degenerate graphs $G$ and (b) all graphs $G$ with $\mbox{mad}(G)<4$, upper bounds on the clique number $\omega(G^2)$ of $G^2$ that match the lower bound given by this construction, up to small additive constants. We show that if $G$ is 2-degenerate with maximum degree $D$, then $\omega(G^2)\le \frac52D+72$ (with $\omega(G^2)\le \frac52D+60$ when $D$ is sufficiently large). And if $G$ has $\mbox{mad}(G)<4$ and maximum degree $D$, then $\omega(G^2)\le \frac52D+532$. Thus, the construction of Hocquard et al. is essentially best possible. Our proofs introduce a "token passing" technique to derive crucial information about non-adjacencies in $G$ of vertices that are adjacent in $G^2$. This is a powerful technique for working with such graphs that has not previously appeared in the literature., Comment: 15 pages, 7 figures; final version incorporates minor reviewer feedback; to appear in J. Graph Theory

Let $G$ be a graph and $k$ be a positive integer, and let $Kc(G, k)$ denote the number of Kempe equivalence classes for the $k$-colorings of $G$. In 2006, Mohar noted that $Kc(G, k) = 1$ if $G$ is bipartite. As a generalization, we show that $Kc(G, k) = 1$ if $G$ is formed from a bipartite graph by adding any number of edges less than $\binom{\lceil k/2\rceil}2+\binom{\lfloor k/2\rfloor}2$. We show that our result is tight (up to lower order terms) by constructing, for each $k \geq 8$, a graph $G$ formed from a bipartite graph by adding $(k^2+8k-45+1)/4$ edges such that $Kc(G, k) \geq 2$. This refutes a recent conjecture of Higashitani--Matsumoto., Comment: 7 pages, 2 figures; 2nd version incorporates reviewer feedback

For a graph $G$ and $t,k\in\mathbb{Z}^+$ a \emph{$t$-tone $k$-coloring} of $G$ is a function $f:V(G)\rightarrow \binom{[k]}{t}$ such that $|f(v)\cap f(w)| < d(v,w)$ for all distinct $v,w \in V(G)$. The \emph{$t$-tone chromatic number} of $G$, denoted $\tau_t(G)$, is the minimum $k$ such that $G$ is $t$-tone $k$-colorable. For small values of $t$, we prove sharp or nearly sharp upper bounds on the $t$-tone chromatic number of various classes of sparse graphs. In particular, we determine $\tau_2(G)$ exactly when $\textrm{mad}(G) < 12/5$ and bound $\tau_2(G)$, up to a small additive constant, when $G$ is outerplanar. We also determine $\tau_t(C_n)$ exactly when $t\in\{3,4,5\}$., Comment: 15 pages, 6 figures, version 2 corrects a few typos

A proper coloring of a graph is \emph{conflict-free} if, for every non-isolated vertex, some color is used exactly once on its neighborhood. Caro, Petru\v{s}evski, and \v{S}krekovski proved that every graph $G$ has a proper conflict-free coloring with at most $5\Delta(G)/2$ colors and conjectured that $\Delta(G)+1$ colors suffice for every connected graph $G$ with $\Delta(G)\ge 3$. Our first main result is that even for list-coloring, $\left\lceil 1.6550826\Delta(G)+\sqrt{\Delta(G)}\right\rceil$ colors suffice for every graph $G$ with $\Delta(G)\ge 10^{8}$; we also prove slightly weaker bounds for all graphs with $\Delta(G)\ge 750$. These results follow from our more general framework on proper conflict-free list-coloring of a pair consisting of a graph $G$ and a "conflict" hypergraph ${\mathcal H}$. As another corollary of our results in this general framework, every graph has a proper $(\sqrt{30}+o(1))\Delta(G)^{1.5}$-list-coloring such that every bi-chromatic component is a path on at most three vertices, where the number of colors is optimal up to a constant factor. Our proof uses a fairly new type of recursive counting argument called Rosenfeld counting, which is a variant of the Lov\'{a}sz Local Lemma or entropy compression. We also prove an asymptotically optimal result for a fractional analogue of our general framework for proper conflict-free coloring for pairs of a graph and a conflict hypergraph. A corollary states that every graph $G$ has a fractional $(1+o(1))\Delta(G)$-coloring such that every fractionally bi-chromatic component has at most two vertices. In particular, it implies that the fractional analogue of the conjecture of Caro et al.\ holds asymptotically in a strong sense.

We survey work on coloring, list coloring, and painting squares of graphs; in particular, we consider strong edge-coloring. We focus primarily on planar graphs and other sparse classes of graphs., Comment: 32 pages, 13 figures and tables, plus 195-entry bibliography, comments are welcome, published as a Dynamic Survey in Electronic Journal of Combinatorics

A \emph{coloring} of a graph $G$ is a map $f:V(G)\to \mathbb{Z}^+$ such that $f(v)\ne f(w)$ for all $vw\in E(G)$. A coloring $f$ is an \emph{odd-sum} coloring if $\sum_{w\in N[v]}f(w)$ is odd, for each vertex $v\in V(G)$. The \emph{odd-sum chromatic number} of a graph $G$, denoted $\chi_{os}(G)$, is the minimum number of colors used (that is, the minimum size of the range) in an odd-sum coloring of $G$. Caro, Petru\v{s}evski, and \v{S}krekovski showed, among other results, that $\chi_{os}(G)$ is well-defined for every finite graph $G$ and, in fact, $\chi_{os}(G)\le 2\chi(G)$. Thus, $\chi_{os}(G)\le 8$ for every planar graph $G$ (by the 4 Color Theorem), $\chi_{os}(G)\le 6$ for every triangle-free planar graph $G$ (by Gr\"{o}tzsch's Theorem), and $\chi_{os}(G)\le 4$ for every bipartite graph. Caro et al. asked, for every even $\Delta\ge 4$, whether there exists $g_{\Delta}$ such that if $G$ is planar with maximum degree $\Delta$ and girth at least $g_{\Delta}$ then $\chi_{os}(G)\le 5$. They also asked, for every even $\Delta\ge 4$, whether there exists $g_{\Delta}$ such that if $G$ is planar and bipartite with maximum degree $\Delta$ and girth at least $g_{\Delta}$ then $\chi_{os}(G)\le 3$. We answer both questions negatively. We also refute a conjecture they made, resolve one further problem they posed, and make progress on another., Comment: 8 pages, 6 figures, to appear in Discrete Applied Math

The coloring reconfiguration graph $\mathcal{C}_k(G)$ has as its vertex set all the proper $k$-colorings of $G$, and two vertices in $\mathcal{C}_k(G)$ are adjacent if their corresponding $k$-colorings differ on a single vertex. Cereceda conjectured that if an $n$-vertex graph $G$ is $d$-degenerate and $k\geq d+2$, then the diameter of $\mathcal{C}_k(G)$ is $O(n^2)$. Bousquet and Heinrich proved that if $G$ is planar and bipartite, then the diameter of $\mathcal{C}_5(G)$ is $O(n^2)$. (This proves Cereceda's Conjecture for every such graph with degeneracy 3.) They also highlighted the particular case of Cereceda's Conjecture when $G$ is planar and has no 3-cycles. As a partial solution to this problem, we show that the diameter of $\mathcal{C}_5(G)$ is $O(n^2)$ for every planar graph $G$ with no 3-cycles and no 5-cycles., Comment: 7 pages, 3 figures, corrects a few errors in the previous version

The reconfiguration graph $\mathcal{C}_k(G)$ for the $k$-colourings of a graph $G$ has a vertex for each proper $k$-colouring of $G$, and two vertices of $\mathcal{C}_k(G)$ are adjacent precisely when those $k$-colourings differ on a single vertex of $G$. Much work has focused on bounding the maximum value of ${\rm{diam}}~\mathcal{C}_k(G)$ over all $n$-vertex graphs $G$. We consider the analogous problems for list colourings and for correspondence colourings. We conjecture that if $L$ is a list-assignment for a graph $G$ with $|L(v)|\ge d(v)+2$ for all $v\in V(G)$, then ${\rm{diam}}~\mathcal{C}_L(G)\le n(G)+\mu(G)$. We also conjecture that if $(L,H)$ is a correspondence cover for a graph $G$ with $|L(v)|\ge d(v)+2$ for all $v\in V(G)$, then ${\rm{diam}}~\mathcal{C}_{(L,H)}(G)\le n(G)+\tau(G)$. (Here $\mu(G)$ and $\tau(G)$ denote the matching number and vertex cover number of $G$.) For every graph $G$, we give constructions showing that both conjectures are best possible. Our first main result proves the upper bounds (for the list and correspondence versions, respectively) ${\rm{diam}}~\mathcal{C}_L(G)\le n(G)+2\mu(G)$ and ${\rm{diam}}~\mathcal{C}_{(L,H)}(G)\le n(G)+2\tau(G)$. Our second main result proves that both conjectured bounds hold, whenever all $v$ satisfy $|L(v)|\ge 2d(v)+1$. We conclude by proving one or both conjectures for various classes of graphs such as complete bipartite graphs, subcubic graphs, cactuses, and graphs with bounded maximum average degree., Comment: 19 pages, 3 figures; to appear in European J. Combinatorics

A proper coloring of a graph is odd if every non-isolated vertex has some color that appears an odd number of times on its neighborhood. This notion was recently introduced by Petru\v{s}evski and \v{S}krekovski, who proved that every planar graph admits an odd $9$-coloring; they also conjectured that every planar graph admits an odd $5$-coloring. Shortly after, this conjecture was confirmed for planar graphs of girth at least seven by Cranston; outerplanar graphs by Caro, Petru\v{s}evski, and \v{S}krekovski. Building on the work of Caro, Petru\v{s}evski, and \v{S}krekovski, Petr and Portier then further proved that every planar graph admits an odd $8$-coloring. In this note we prove that every 1-planar graph admits an odd $23$-coloring, where a graph is 1-planar if it can be drawn in the plane so that each edge is crossed by at most one other edge., Comment: Third version improves upper bound from 31 to 23

Fix a graph $G$, a list-assignment $L$ for $G$, and $L$-colorings $\alpha$ and $\beta$. An $L$-recoloring sequence, starting from $\alpha$, recolors a single vertex at each step, so that each resulting intermediate coloring is a proper $L$-coloring. An $L$-recoloring sequence transforms $\alpha$ to $\beta$ if its initial coloring is $\alpha$ and its final coloring is $\beta$. We prove there exists an $L$-recoloring sequence that transforms $\alpha$ to $\beta$ and recolors each vertex at most a constant number of times if (i) $G$ is triangle-free and planar and $L$ is a 7-assignment, or (ii) $\mathrm{mad}(G)<17/5$ and $L$ is a 6-assignment or (iii) $\mathrm{mad}(G)<22/9$ and $L$ is a 4-assignment. Parts (i) and (ii) confirm conjectures of Dvo\v{r}\'{a}k and Feghali., Comment: 11 pages, 5 figures; to appear in European J. Combinatorics

A proper coloring of a graph is called \emph{odd} if every non-isolated vertex has some color that appears an odd number of times on its neighborhood. The smallest number of colors that admits an odd coloring of a graph $G$ is denoted $\chi_o(G)$. This notion was introduced by Petru\v{s}evski and \v{S}krekovski, who proved that if $G$ is planar then $\chi_o(G)\le 9$; they also conjectured that $\chi_o(G)\le 5$. For a positive real number $\alpha$, we consider the maximum value of $\chi_o(G)$ over all graphs $G$ with maximum average degree less than $\alpha$; we denote this value by $\chi_o(\mathcal{G}_{\alpha})$. We note that $\chi_o(\mathcal{G}_{\alpha})$ is undefined for all $\alpha\ge 4$. In contrast, for each $\alpha\in[0,4)$, we give a (nearly sharp) upper bound on $\chi_o(\mathcal{G}_{\alpha})$. Finally, we prove $\chi_o(\mathcal{G}_{20/7})= 5$ and $\chi_o(\mathcal{G}_3)= 6$. Both of these results are sharp., Comment: 8 pages

An $\alpha,\beta$-Kempe swap in a properly colored graph interchanges the colors on some component of the subgraph induced by colors $\alpha$ and $\beta$. Two $k$-colorings of a graph are $k$-Kempe equivalent if we can form one from the other by a sequence of Kempe swaps (never using more than $k$ colors). Las Vergnas and Meyniel showed that if a graph is $(k-1)$-degenerate, then each pair of its $k$-colorings are $k$-Kempe equivalent. Mohar conjectured the same conclusion for connected $k$-regular graphs. This was proved for $k=3$ by Feghali, Johnson, and Paulusma (with a single exception $K_2\square K_3$, also called the 3-prism) and for $k\ge 4$ by Bonamy, Bousquet, Feghali, and Johnson. In this paper we prove an analogous result for list-coloring. For a list-assignment $L$ and an $L$-coloring $\varphi$, a Kempe swap is called $L$-valid for $\varphi$ if performing the Kempe swap yields another $L$-coloring. Two $L$-colorings are called $L$-equivalent if we can form one from the other by a sequence of $L$-valid Kempe swaps. Let $G$ be a connected $k$-regular graph with $k\ge 3$. We prove that if $L$ is a $k$-assignment, then all $L$-colorings are $L$-equivalent (again with a single exception $K_2 \square K_3$). When $k\ge 4$, the proof is completely self-contained, so implies an alternate proof of the result of Bonamy et al. Our proofs rely on the following key lemma, which may be of independent interest. Let $H$ be a graph such that for every degree-assignment $L_H$ all $L_H$-colorings are $L_H$-equivalent. If $G$ is a connected graph that contains $H$ as an induced subgraph, then for every degree-assignment $L_G$ for $G$ all $L_G$-colorings are $L_G$-equivalent., Comment: 29 pages, 12 figures; second version extends the main result to cliques, which were previously excluded; third version incorporates reviewer feedback; to appear in Combinatorica

A strong edge-coloring $\varphi$ of a graph $G$ assigns colors to edges of $G$ such that $\varphi(e_1)\ne \varphi(e_2)$ whenever $e_1$ and $e_2$ are at distance no more than 1. It is equivalent to a proper vertex coloring of the square of the line graph of $G$. In 1990 Faudree, Schelp, Gy\'arf\'as, and Tuza conjectured that if $G$ is a bipartite graph with maximum degree 3 and sufficiently large girth, then $G$ has a strong edge-coloring with at most 5 colors. In 2021 this conjecture was disproved by Lu\v{z}ar, Ma\v{c}ajov\'{a}, \v{S}koviera, and Sot\'{a}k. Here we give an alternative construction to disprove the conjecture., Comment: 3.5 pages, 1 figure; second version extends the construction from the 3-regular case to the $k$-regular case, for each integer $k\ge 2$; third version incorporates referee feedback; to appear in Discrete Applied Math

A set of geometric graphs is {\em geometric-packable} if it can be asymptotically packed into every sequence of drawings of the complete graph $K_n$. For example, the set of geometric triangles is geometric-packable due to the existence of Steiner Triple Systems. When $G$ is the $4$-cycle (or $4$-cycle with a chord), we show that the set of plane drawings of $G$ is geometric-packable. In contrast, the analogous statement is false when $G$ is nearly any other planar Hamiltonian graph (with at most 3 possible exceptions). A convex geometric graph is {\em convex-packable} if it can be asymptotically packed into the convex drawings of the complete graphs. For each planar Hamiltonian graph $G$, we determine whether or not a plane $G$ is convex-packable. Many of our proofs explicitly construct these packings; in these cases, the packings exhibit a symmetry that mirrors the vertex transitivity of $K_n$.

For a list assignment $L$ and an $L$-coloring $\varphi$, a Kempe swap in $\varphi$ is \emph{$L$-valid} if it yields another $L$-coloring. Two $L$-colorings are \emph{$L$-equivalent} if we can form one from another by a sequence of $L$-valid Kempe swaps. And a graph $G$ is \emph{$L$-swappable} if every two of its $L$-colorings are $L$-equivalent. We consider $L$-swappability of line graphs of planar graphs with large maximum degree. Let $G$ be a planar graph with $\Delta(G)\ge 9$ and let $H$ be the line graph of $G$. If $L$ is a $(\Delta(G)+1)$-assignment to $H$, then $H$ is $L$-swappable. Let $G$ be a planar graph with $\Delta(G)\ge 15$ and let $H$ be the line graph of $G$. If $L$ is a $\Delta(G)$-assignment to $H$, then $H$ is $L$-swappable. The first result is analogous to one for $L$-choosability by Borodin, which was later strengthened by Bonamy. The second result is analogous to another for $L$-choosability by Borodin, which was later strengthened by Borodin, Kostochka, and Woodall., Comment: 15 pages, 5 figures, 2 page appendix; to appear in Discrete Math (special issue in honor of Landon Rabern)

The planar Turan number $\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$ is the largest number of edges in an $n$-vertex planar graph with no $\ell$-cycle. For $\ell\in \{3,4,5,6\}$, upper bounds on $\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$ are known that hold with equality infinitely often. Ghosh, Gy\"{o}ri, Martin, Paulo, and Xiao [arxiv:2004.14094] conjectured an upper bound on $\textrm{ex}_{\mathcal{P}}(C_{\ell},n)$ for every $\ell\ge 7$ and $n$ sufficiently large. We disprove this conjecture for every $\ell\ge 11$. We also propose two revised versions of the conjecture., Comment: 9 pages, 2 figures

A proper coloring of the vertices of a graph is called a \emph{star coloring} if the union of every two color classes induces a star forest. The star chromatic number $\chi_s(G)$ is the smallest number of colors required to obtain a star coloring of $G$. In this paper, we study the relationship between the star chromatic number $\chi_s(G)$ and the maximum average degree $\mbox{Mad}(G)$ of a graph $G$. We prove that: (1) If $G$ is a graph with $\mbox{Mad}(G) < \frac{26}{11}$, then $\chi_s(G)\leq 4$. (2) If $G$ is a graph with $\mbox{Mad}(G) < \frac{18}{7}$ and girth at least 6, then $\chi_s(G)\leq 5$. (3) If $G$ is a graph with $\mbox{Mad}(G) < \frac{8}{3}$ and girth at least 6, then $\chi_s(G)\leq 6$. These results are obtained by proving that such graphs admit a particular decomposition into a forest and some independent sets., Comment: 20 pages, 5 figures

A Kempe swap in a proper coloring interchanges the colors on some maximal connected 2-colored subgraph. Two $k$-colorings are $k$-equivalent if we can transform one into the other using Kempe swaps. The triangulated toroidal grid, $T[m\times n]$, is formed from (a toroidal embedding of) the Cartesian product of $C_m$ and $C_n$ by adding parallel diagonals inside all 4-faces. Mohar and Salas showed that not all 4-colorings of $T[m\times n]$ are 4-equivalent. In contrast, Bonamy, Bousquet, Feghali, and Johnson showed that all 6-colorings of $T[m\times n]$ are 6-equivalent. They asked whether the same is true for 5-colorings. We answer their question affirmatively when $m,n\ge 6$. Further, we show that if $G$ is 6-regular with a toroidal embedding where every non-contractible cycle has length at least 7, then all 5-colorings of $G$ are 5-equivalent. Our results relate to the antiferromagnetic Pott's model in statistical mechanics., Comment: 17 pages, 16 figures; 4th version incorporates minor referee feedback; to appear in European Journal of Combinatorics

For each integer k >= 2, we determine a sharp bound on mad(G) such that V(G) can be partitioned into sets I and F_k, where I is an independent set and G[F_k] is a forest in which each component has at most k vertices. For each k we construct an infinite family of examples showing our result is best possible. Our results imply that every planar graph G of girth at least 9 (resp. 8, 7) has a partition of V(G) into an independent set I and a set F such that G[F] is a forest with each component of order at most 3 (resp. 4, 6). Hendrey, Norin, and Wood asked for the largest function g(a,b) such that if mad(G) < g(a,b) then V(G) has a partition into sets A and B such that mad(G[A]) < a and mad(G[B]) < b. They specifically asked for the value of g(1,b), i.e., the case when A is an independent set. Previously, the only values known were g(1,4/3) and g(1,2). We find g(1,b) whenever 4/3 < b < 2., Comment: 21 pages, 9 figures and 2 tables; this version incorporates reviewer feedback; to appear in SIAM J. Discrete Math

The Borodin-Kostochka Conjecture states that for a graph $G$, if $\Delta(G) \geq 9$ and $\omega(G) \leq \Delta(G)-1$, then $\chi(G)\leq\Delta(G) -1$. We prove the Borodin-Kostochka Conjecture for $(P_5, \text{gem})$-free graphs, i.e., graphs with no induced $P_5$ and no induced $K_1\vee P_4$., Comment: 8 pages, 6 figures

We introduce a new directed graph model for social networks, based on the transitivity of triads. In the Iterated Local Directed Transitivity (ILDT) model, new nodes are born over discrete time-steps, and inherit the link structure of their parent nodes. The ILDT model may be viewed as a directed analogue of the ILT model for undirected graphs introduced in \cite{ilt}. We investigate network science and graph theoretical properties of ILDT digraphs. We prove that the ILDT model exhibits a densification power law, so that the digraphs generated by the models densify over time. The number of directed triads are investigated, and counts are given of the number of directed 3-cycles and transitive $3$-cycles. A higher number of transitive 3-cycles are generated by the ILDT model, as found in real-world, on-line social networks. In many instances of the chosen initial digraph, the model eventually generates graphs with Hamiltonian directed cycles. We finish with a discussion of the eigenvalues of the adjacency matrices of ILDT directed graphs, and provide further directions.

A multigraph $G$ is near-bipartite if $V(G)$ can be partitioned as $I,F$ such that $I$ is an independent set and $F$ induces a forest. We prove that a multigraph $G$ is near-bipartite when $3|W|-2|E(G[W])|\ge -1$ for every $W\subseteq V(G)$, and $G$ contains no $K_4$ and no Moser spindle. We prove that a simple graph $G$ is near-bipartite when $8|W|-5|E(G[W])|\ge -4$ for every $W\subseteq V(G)$, and $G$ contains no subgraph from some finite family $\mathcal{H}$. We also construct infinite families to show that both results are best possible in a very sharp sense., Comment: 37 pages, 21 figures; incorporates reviewer feedback; to appear in SIAM J. Discrete Math

For integers $a\ge 2b>0$, a \emph{circular $a/b$-flow} is a flow that takes values from $\{\pm b, \pm(b+1), \dots, \pm(a-b)\}$. The Planar Circular Flow Conjecture states that every $2k$-edge-connected planar graph admits a circular $(2+\frac{2}{k})$-flow. The cases $k=1$ and $k=2$ are equivalent to the Four Color Theorem and Gr\"otzsch's 3-Color Theorem. For $k\ge 3$, the conjecture remains open. Here we make progress when $k=4$ and $k=6$. We prove that (i) {\em every 10-edge-connected planar graph admits a circular 5/2-flow} and (ii) {\em every 16-edge-connected planar graph admits a circular 7/3-flow.} The dual version of statement (i) on circular coloring was previously proved by Dvo\v{r}\'ak and Postle (Combinatorica 2017), but our proof has the advantages of being much shorter and avoiding the use of computers for case-checking. Further, it has new implications for antisymmetric flows. Statement (ii) is especially interesting because the counterexamples to Jaeger's original Circular Flow Conjecture are 12-edge-connected nonplanar graphs that admit no circular 7/3-flow. Thus, the planarity hypothesis of (ii) is essential., Comment: 18 pages, 6 figures, plus 2.5 page appendix with 2 figures, comments welcome

We prove several results on coloring squares of planar graphs without 4-cycles. First, we show that if $G$ is such a graph, then $G^2$ is $(\Delta(G)+72)$-degenerate. This implies an upper bound of $\Delta(G)+73$ on the chromatic number of $G^2$ as well as on several variants of the chromatic number such as the list-chromatic number, paint number, Alon--Tarsi number, and correspondence chromatic number. We also show that if $\Delta(G)$ is sufficiently large, then the upper bounds on each of these parameters of $G^2$ can all be lowered to $\Delta(G)+2$ (which is best possible). To complement these results, we show that 4-cycles are unique in having this property. Specifically, let $S$ be a finite list of positive integers, with $4\notin S$. For each constant $C$, we construct a planar graph $G_{S,C}$ with no cycle with length in $S$, but for which $\chi(G_{S,C}^2) > \Delta(G_{S,C})+C$., Comment: 24 pages, 6 figures; revised version incorporates suggestions of referees; to appear in Combinatorica

In \emph{$k$-bootstrap percolation}, we fix $p\in (0,1)$, an integer $k$, and a plane graph $G$. Initially, we infect each face of $G$ independently with probability $p$. Infected faces remain infected forever, and if a healthy (uninfected) face has at least $k$ infected neighbors, then it becomes infected. For fixed $G$ and $p$, the \emph{percolation threshold} is the largest $k$ such that eventually all faces become infected, with probability at least $1/2$. For a large class of infinite graphs, we show that this threshold is independent of $p$. We consider bootstrap percolation in tilings of the plane by regular polygons. A \emph{vertex type} in such a tiling is the cyclic order of the faces that meet a common vertex. First, we determine the percolation threshold for each of the Archimedean lattices. More generally, let $\mathcal{T}$ denote the set of plane tilings $T$ by regular polygons such that if $T$ contains one instance of a vertex type, then $T$ contains infinitely many instances of that type. We show that no tiling in $\mathcal{T}$ has threshold 4 or more. Further, the only tilings in $\mathcal{T}$ with threshold 3 are four of the Archimedean lattices. Finally, we describe a large subclass of $\mathcal{T}$ with threshold 2., Comment: 10 pages, 5 figures; to appear in Australasian J. Combinatorics; this version incorporates reviewer and editor feedback, and differs from the final journal version mainly in typesetting

A graph $G$ is \emph{$(a,b)$-choosable} if given any list assignment $L$ with $|L(v)|=a$ for each $v\in V(G)$ there exists a function $\varphi$ such that $\varphi(v)\in L(v)$ and $|\varphi(v)|=b$ for all $v\in V(G)$, and whenever vertices $x$ and $y$ are adjacent $\varphi(x)\cap \varphi(y)=\emptyset$. Meng, Puleo, and Zhu conjectured a characterization of (4,2)-choosable graphs. We prove their conjecture., Comment: 20 pages, 20 figures; version 3 incorporates reviewer feedback: completely rewrote Section 5 to correct an error, omitted many tedious details of showing that certain graphs are not (4,2)-choosable, removed open question and conjecture; to appear in J. Graph Theory

The distinguishing chromatic number, $\chi_D(G)$, of a graph $G$ is the smallest number of colors in a proper coloring, $\varphi$, of $G$, such that the only automorphism of $G$ that preserves all colors of $\varphi$ is the identity map. Collins and Trenk conjectured that if $G$ is connected with girth at least 5 and $G\ne C_6$, then $\chi_D(G)\le \Delta+1$. We prove this conjecture., Comment: 9 pages, 3 figures; expanded introduction and bibliography from version 1

An \emph{acyclic edge-coloring} of a graph $G$ is a proper edge-coloring of $G$ such that the subgraph induced by any two color classes is acyclic. The \emph{acyclic chromatic index}, $\chi'_a(G)$, is the smallest number of colors allowing an acyclic edge-coloring of $G$. Clearly $\chi'_a(G)\ge \Delta(G)$ for every graph $G$. Cohen, Havet, and M\"{u}ller conjectured that there exists a constant $M$ such that every planar graph with $\Delta(G)\ge M$ has $\chi'_a(G)=\Delta(G)$. We prove this conjecture., Comment: 15 pages, 8 figures; minor revisions to incorporate reviewer feedback; to appear in SIAM J. Discrete Math

A simple graph $G$ is \emph{overfull} if $|E(G)|>\Delta\lfloor|V(G)|/2\rfloor$. By the pigeonhole principle, every overfull graph $G$ has $\chi'(G)>\Delta$. The \emph{core} of a graph, denoted $G_\Delta$, is the subgraph induced by its vertices of degree $\Delta$. Vizing's Adjacency Lemma implies that if $\chi'(G)>\Delta$, then $G_\Delta$ contains cycles. Hilton and Zhao conjectured that if $G_\Delta$ has maximum degree 2 and $\Delta\ge 4$, then $\chi'(G)>\Delta$ precisely when $G$ is overfull. We prove this conjecture for the case $\Delta=4$., Comment: 13 pages, 9 figures; this version incorporates reviewer feedback and corrects a few errors; to appear in SIAM J. Discrete Math

In 2011, the second author conjectured that every line graph $G$ satisfies $\chi(G)\le \max\{\omega(G),\frac{5\Delta(G)+8}{6}\}$. This conjecture is best possible, as shown by replacing each edge in a 5-cycle by $k$ parallel edges, and taking the line graph. In this paper we prove the conjecture. We also develop more general techniques and results that will likely be of independent interest, due to their use in attacking the Goldberg--Seymour conjecture., Comment: 28 pages, 5 figures

The 4 Color Theorem (4CT) implies that every $n$-vertex planar graph has an independent set of size at least $\frac{n}4$; this is best possible, as shown by the disjoint union of many copies of $K_4$. In 1968, Erd\H{o}s asked whether this bound on independence number could be proved more easily than the full 4CT. In 1976 Albertson showed (independently of the 4CT) that every $n$-vertex planar graph has an independent set of size at least $\frac{2n}9$. Until now, this remained the best bound independent of the 4CT. Our main result improves this bound to $\frac{3n}{13}$., Comment: 25 pages, 9 figures

A graph $G$ is $k$-critical if $G$ is not $(k-1)$-colorable, but every proper subgraph of $G$ is $(k-1)$-colorable. A graph $G$ is $k$-choosable if $G$ has an $L$-coloring from every list assignment $L$ with $|L(v)|=k$ for all $v$, and a graph $G$ is \emph{$k$-list-critical} if $G$ is not $(k-1)$-choosable, but every proper subgraph of $G$ is $(k-1)$-choosable. The problem of bounding (from below) the number of edges in a $k$-critical graph has been widely studied, starting with work of Gallai and culminating with the seminal results of Kostochka and Yancey, who essentially solved the problem. In this paper, we improve the best lower bound on the number of edges in a $k$-list-critical graph. Our proof uses the discharging method, which makes it simpler and more modular than previous work in this area., Comment: 17 pages, 2 figures

Let $G$ be a connected graph with maximum degree $\Delta$. Brooks' theorem states that $G$ has a $\Delta$-coloring unless $G$ is a complete graph or an odd cycle. A graph $G$ is \emph{degree-choosable} if $G$ can be properly colored from its lists whenever each vertex $v$ gets a list of $d(v)$ colors. In the context of list coloring, Brooks' theorem can be strengthened to the following. Every connected graph $G$ is degree-choosable unless each block of $G$ is a complete graph or an odd cycle; such a graph $G$ is a \emph{Gallai tree}. This degree-choosability result was further strengthened to Alon--Tarsi orientations; these are orientations of $G$ in which the number of spanning Eulerian subgraphs with an even number of edges differs from the number with an odd number of edges. A graph $G$ is \emph{degree-AT} if $G$ has an Alon--Tarsi orientation in which each vertex has indegree at least 1. Alon and Tarsi showed that if $G$ is degree-AT, then $G$ is also degree-choosable. Hladky, Kral, and Schauz showed that a connected graph is degree-AT if and only if it is not a Gallai tree. In this paper, we consider pairs $(G,x)$ where $G$ is a connected graph and $x$ is some specified vertex in $V(G)$. We characterize pairs such that $G$ has no Alon--Tarsi orientation in which each vertex has indegree at least 1 and $x$ has indegree at least 2. When $G$ is 2-connected, the characterization is simple to state., Comment: 14 pages, 4 figures

Given a configuration of pebbles on the vertices of a connected graph $G$, a \emph{pebbling move} removes two pebbles from some vertex and places one pebble on an adjacent vertex. The \emph{pebbling number} of a graph $G$ is the smallest integer $k$ such that for each vertex $v$ and each configuration of $k$ pebbles on $G$ there is a sequence of pebbling moves that places at least one pebble on $v$. First, we improve on results of Hurlbert, who introduced a linear optimization technique for graph pebbling. In particular, we use a different set of weight functions, based on graphs more general than trees. We apply this new idea to some graphs from Hurlbert's paper to give improved bounds on their pebbling numbers. Second, we investigate the structure of Class 0 graphs with few edges. We show that every $n$-vertex Class 0 graph has at least $\frac53n - \frac{11}3$ edges. This disproves a conjecture of Blasiak et al. For diameter 2 graphs, we strengthen this lower bound to $2n - 5$, which is best possible. Further, we characterize the graphs where the bound holds with equality and extend the argument to obtain an identical bound for diameter 2 graphs with no cut-vertex., Comment: 19 pages, 8 figures