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The $(n-\ell)$-deck of an $n$-vertex graph is the multiset of subgraphs obtained from it by deleting $\ell$ vertices. A family of $n$-vertex graphs is $\ell$-recognizable if every graph having the same $(n-\ell)$-deck as a graph in the family is also in the family. We prove that the family of $n$-vertex graphs with no cycles is $\ell$-recognizable when $n\ge2\ell+1$ (except for $(n,\ell)=(5,2)$). As a consequence, the family of $n$-vertex trees is $\ell$-recognizable when $n\ge2\ell+1$ and $\ell\ne2$. It is known that this fails when $n=2\ell$.

The $(n-\ell)$-deck of an $n$-vertex graph is the multiset of (unlabeled) subgraphs obtained from it by deleting $\ell$ vertices. An $n$-vertex graph is $\ell$-reconstructible if it is determined by its $(n-\ell)$-deck, meaning that no other graph has the same deck. We prove that every tree with at least $6\ell+11$ vertices is $\ell$-reconstructible.

Given a nonnegative integer weight $f(v)$ for each vertex $v$ in a multigraph $G$, an {\it $f$-bounded subgraph} of $G$ is a multigraph $H$ contained in $G$ such that $d_H(v)\le f(v)$ for all $v\in V(G)$. Using Tutte's $f$-Factor Theorem, we give a new proof of the min-max relation for the maximum size of an $f$-bounded subgraph of $G$. When $f(v)=1$ for all $v$, the formula reduces to the classical Tutte--Berge Formula for the maximum size of a matching., Comment: 7 pages

The \emph{eccentricity} of a vertex $u$ in a graph $G$, denoted by $e_G(u)$, is the maximum distance from $u$ to other vertices in $G$. We study extremal problems for the average eccentricity and the first and second Zagreb eccentricity indices, denoted by $\sigma_0(G)$, $\sigma_1(G)$, and $\sigma_2(G)$, respectively. These are defined by $\sigma_0(G)=\frac{1}{|V(G)|}\sum_{u\in V(G)}e_G(u)$, $\sigma_1(G)=\sum_{u\in V(G)}e_G^2(u)$, and $\sigma_2(G)=\sum_{uv\in E(G)}e_G(u)e_G(v)$. We study lower and upper bounds on these parameters among $n$-vertex connected graphs with fixed diameter, chromatic number, clique number, or matching number. Most of the bounds are sharp, with the corresponding extremal graphs characterized., Comment: 27 pages

We study the minimum number of maximum matchings in a bipartite multigraph G with parts $X$ and $Y$ under various conditions, refining the well-known lower bound due to M. Hall. When $|X|=n$, every vertex in $X$ has degree at least $k$, and every vertex in $X$ has at least $r$ distinct neighbors, the minimum is $r!(k-r+1)$ when $n\ge r$ and is $[r+n(k-r)]\prod_{i=1}^{n-1}(r-i)$ when $n

A graph is $\ell$-reconstructible if it is determined by its multiset of induced subgraphs obtained by deleting $\ell$ vertices. We prove that strongly regular graphs with at least six vertices are $2$-reconstructible., Comment: 5 pages

Visibility representation of digraphs was introduced by Axenovich, Beveridge, Hutch\-inson, and West (\emph{SIAM J. Discrete Math.} {\bf 27}(3) (2013) 1429--1449) as a natural generalization of $t$-bar visibility representation of undirected graphs. A {\it $t$-bar visibility representation} of a digraph $G$ assigns each vertex at most $t$ horizontal bars in the plane so that there is an arc $xy$ in the digraph if and only if some bar for $x$ "sees" some bar for $y$ above it along an unblocked vertical strip with positive width. The {\it visibility number} $b(G)$ is the least $t$ such that $G$ has a $t$-bar visibility representation. In this paper, we solve several problems about $b(G)$ posed by Axenovich et al.\ and prove that determining whether the bar visibility number of a digraph is $2$ is NP-complete., Comment: 17 pages,9 figures

The $(n-\ell)$-deck of an $n$-vertex graph is the multiset of subgraphs obtained from it by deleting $\ell$ vertices. A family of $n$-vertex graphs is $\ell$-recognizable if every graph having the same $(n-\ell)$-deck as a graph in the family is also in the family. We prove that the family of $n$-vertex graphs having no cycles is $\ell$-recognizable when $n\ge2\ell+1$ (except for $(n,\ell)=(5,2)$). It is known that this fails when $n=2\ell$., Comment: 16 pages

The Reconstruction Conjecture of Ulam asserts that, for $n\geq 3$, every $n$-vertex graph is determined by the multiset of its induced subgraphs with $n-1$ vertices. The conjecture is known to hold for various special classes of graphs but remains wide open. We survey results on the more general conjecture by Kelly from 1957 that for every positive integer $\ell$ there exists $M_\ell$ (with $M_1=3$) such that when $n\geq M_\ell$ every $n$-vertex graph is determined by the multiset of its induced subgraphs with $n-\ell$ vertices.

The 1970s conjecture of Lov\'asz and Plummer that the number of perfect matchings in any $3$-regular graph is exponential in the number of vertices was proved in 2011 by Esperet, Kardo\v{s}, King, Kr\'al', and Norine. We give the exact formula for the number of perfect matchings in two families of $3$-regular graphs. In the graph consisting of a $2n$-cycle with diametric chords (also known as the M\"obius ladder $M_n$ and a Harary graph) and in the cartesian product of the cycle $C_n$ with an edge (called the cycle prism), the number of matchings is the sum of the Fibonacci numbers $F_{n-1}$ and $F_{n+1}$, plus two more for the M\"obius ladder when $n$ is odd and for the cycle prism when $n$ is even., Comment: The authors have been informed that the results in this paper are not new; the sequences for the two problems appear in http://oeis.org/A020878 and http://oeis.org/A068397 , respectively. The results can also be found in J.P. McSorley, Counting structures in the M\"obius ladder, Discrete Math. 184 (1998), no. 1-3, 137--164. The result for the prism was known to E. Deutsch in 2004 (unpublished)

Tuza [1992] proved that a graph with no cycles of length congruent to $1$ modulo $k$ is $k$-colorable. We prove that if a graph $G$ has an edge $e$ such that $G-e$ is $k$-colorable and $G$ is not, then for $2\leq r\leq k$, the edge $e$ lies in at least $\prod_{i=1}^{r-1}(k-i)$ cycles of length $1\mod r$ in $G$, and $G-e$ contains at least $\frac{1}{2}\prod_{i=1}^{r-1}(k-i)$ cycles of length $0 \mod r$. A $(k,d)$-coloring of $G$ is a homomorphism from $G$ to the graph $K_{k:d}$ with vertex set $\mathbb{Z}_{k}$ defined by making $i$ and $j$ adjacent if $d\leq j-i \leq k-d$. When $k$ and $d$ are relatively prime, define $s$ by $sd\equiv 1\mod k$. A result of Zhu [2002] implies that $G$ is $(k,d)$-colorable when $G$ has no cycle $C$ with length congruent to $is$ modulo $k$ for any $i\in \{1,\ldots,2d-1\}$. In fact, only $d$ classes need be excluded: we prove that if $G-e$ is $(k,d)$-colorable and $G$ is not, then $e$ lies in at least one cycle with length congruent to $is\mod k$ for some $i$ in $\{1,\ldots,d\}$. Furthermore, if this does not occur with $i\in\{1,\ldots,d-1\}$, then $e$ lies in at least two cycles with length $1\mod k$ and $G-e$ contains a cycle of length $0 \mod k$., Comment: 10 pages

A graph is $\ell$-reconstructible if it is determined by its multiset of induced subgraphs obtained by deleting $\ell$ vertices. We prove that $3$-regular graphs are $2$-reconstructible.

A $t$-bar visibility representation of a graph assigns each vertex up to $t$ horizontal bars in the plane so that two vertices are adjacent if and only if some bar for one vertex can see some bar for the other via an unobstructed vertical channel of positive width. The least $t$ such that $G$ has a $t$-bar visibility representation is the bar visibility number of $G$, denoted by $b(G)$. For the complete bipartite graph $K_{m,n}$, the lower bound $b(K_{m,n})\ge\lceil{\frac{mn+4}{2m+2n}}\rceil$ from Euler's Formula is well known. We prove that equality holds., Comment: 16 pages, 6 figures

The $k$-deck of a graph is the multiset of its subgraphs induced by $k$ vertices. A graph or graph property is $l$-reconstructible if it is determined by the deck of subgraphs obtained by deleting $l$ vertices. We show that the degree list of an $n$-vertex graph is $3$-reconstructible when $n\ge7$, and the threshold on $n$ is sharp. Using this result, we show that when $n\ge7$ the $(n-3)$-deck also determines whether an $n$-vertex graph is connected; this is also sharp. These results extend the results of Chernyak and Manvel, respectively, that the degree list and connectedness are $2$-reconstructible when $n\ge6$, which are also sharp., Comment: 12 pages

A $t$-bar visibility representation of a graph assigns each vertex up to $t$ horizontal bars in the plane so that two vertices are adjacent if and only if some bar for one vertex can see some bar for the other via an unobstructed vertical channel of positive width. The least $t$ such that $G$ has a $t$-bar visibility representation is the bar visibility number of $G$, denoted by $b(G)$. We show that if $H$ is a spanning subgraph of $G$, then $b(H)\le b(G)+1$. It follows that $b(G)\le \lceil n/6\rceil+1$ when $G$ is an $n$-vertex graph. This improves the upper bound obtained by Chang et al. (SIAM J. Discrete Math. 18 (2004) 462)., Comment: 6 pages,1 figure

For a graph $G$, let $f_2(G)$ denote the largest number of vertices in a $2$-regular subgraph of $G$. We determine the minimum of $f_2(G)$ over $3$-regular $n$-vertex simple graphs $G$. To do this, we prove that every $3$-regular multigraph with exactly $c$ cut-edges has a $2$-regular subgraph that omits at most $\max\{0,\lfloor (c-1)/2\rfloor\}$ vertices. More generally, every $n$-vertex multigraph with maximum degree $3$ and $m$ edges has a $2$-regular subgraph that omits at most $\max\{0,\lfloor (3n-2m+c-1)/2\rfloor\}$ vertices. These bounds are sharp; we describe the extremal multigraphs.

An ordered hypergraph is a hypergraph $H$ with a specified linear ordering of the vertices, and the appearance of an ordered hypergraph $G$ in $H$ must respect the specified order on $V(G)$. In on-line Ramsey theory, Builder iteratively presents edges that Painter must immediately color. The $t$-color on-line size Ramsey number $\tilde R_t (G)$ of an ordered hypergraph $G$ is the minimum number of edges Builder needs to play (on a large ordered set of vertices) to force Painter using $t$ colors to produce a monochromatic copy of $G$. The monotone tight path $P_r^{(k)}$ is the ordered hypergraph with $r$ vertices whose edges are all sets of $k$ consecutive vertices. We obtain good bounds on $\tilde R_t (P_r^{(k)})$. Letting $m=r-k+1$ (the number of edges in $P_r^{(k)}$), we prove $m^{t-1}/(3\sqrt t)\le\tilde R_t (P_r^{(2)})\le tm^{t+1}$. For general $k$, a trivial upper bound is ${R \choose k}$, where $R$ is the least number of vertices in a $k$-uniform (ordered) hypergraph whose $t$-colorings all contain $P_r^{(k)}$ (and is a tower of height $k-2$). We prove $R/(k\lg R)\le\tilde R_t(P_r^{(k)})\le R(\lg R)^{2+\epsilon}$, where $\epsilon$ is any positive constant and $t(m-1)$ is sufficiently large. Our upper bounds improve prior results when $t$ grows faster than $m/\log m$. We also generalize our results to $\ell$-loose monotone paths, where each successive edge begins $\ell$ vertices after the previous edge.

We study $2k$-factors in $(2r+1)$-regular graphs. Hanson, Loten, and Toft proved that every $(2r+1)$-regular graph with at most $2r$ cut-edges has a $2$-factor. We generalize their result by proving for $k\le(2r+1)/3$ that every $(2r+1)$-regular graph with at most $2r-3(k-1)$ cut-edges has a $2k$-factor. Both the restriction on $k$ and the restriction on the number of cut-edges are sharp. We characterize the graphs that have exactly $2r-3(k-1)+1$ cut-edges but no $2k$-factor. For $k>(2r+1)/3$, there are graphs without cut-edges that have no $2k$-factor, as studied by Bollob\'as, Saito, and Wormald., Comment: 9 pages

An ordered graph $G$ is a graph together with a specified linear ordering on the vertices, and its interval chromatic number is the minimum number of independent sets consisting of consecutive vertices that are needed to partition the vertex set. The $t$-color Ramsey number $R_t(G)$ of an ordered graph $G$ is the minimum number of vertices of an ordered complete graph such that every edge-coloring from a set of $t$ colors contains a monochromatic copy of $G$ such that the copy of $G$ preserves the original ordering on $G$. An ordered graph is $k$-ichromatic if it has interval chromatic number $k$. We obtain lower bounds linear in the number of vertices for the Ramsey numbers of certain classes of 2-ichromatic ordered graphs. We also determine the exact value of the $t$-color Ramsey number for two families of 2-ichromatic ordered graphs, and we prove a linear upper bound for a class of 2-ichromatic ordered graphs., Comment: 13 pages

The \emph{slow-coloring game} is played by Lister and Painter on a graph $G$. Initially, all vertices of $G$ are uncolored. In each round, Lister marks a nonempty set $M$ of uncolored vertices, and Painter colors a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. The score of the game is the sum of the sizes of all sets marked by Lister. The goal of Painter is to minimize the score, while Lister tries to maximize it. We provide strategies for Painter on various classes of graphs whose vertices can be partitioned into a bounded number of sets inducing forests, including $k$-degenerate, acyclically $k$-colorable, planar, and outerplanar graphs. For example, we show that on an $n$-vertex graph $G$, Painter can keep the score to at most $\frac{3k+4}4n$ when $G$ is $k$-degenerate, $3.9857n$ when $G$ is acyclically $5$-colorable, $3n$ when $G$ is planar with a Hamiltonian dual, $\frac{8n+3m}5$ when $G$ is $4$-colorable with $m$ edges (hence $3.4n$ when $G$ is planar), and $\frac73n$ when $G$ is outerplanar., Comment: 15 pages, 3 figures

Let $G$ be a graph with nonnegative integer weights. A {\it unit acquisition move} transfers one unit of weight from a vertex to a neighbor that has at least as much weight. The {\it unit acquisition number} of a graph $G$, denoted $a_u(G)$, is the minimum size that the set of vertices with positive weight can be reduced to via successive unit acquisition moves when starting from the configuration in which every vertex has weight $1$. For a graph $G$ with $n$ vertices and minimum degree $k$, we prove $a_u(G)\le (n-1)/k$, with equality for complete graphs and $C_5$. Also $a_u(G)$ is at most the minimum size of a maximal matching in $G$, with equality on an infinite family of graphs. Furthermore, $a_u(G)$ is bounded by the maximum degree and by $\sqrt{n-1}$ when $G$ is an $n$-vertex tree with diameter at most $4$. We also construct arbitrarily large trees with maximum degree $5$ having unit acquisition number $1$, obtain a linear-time algorithm to compute the acquisition number of a caterpillar, and show that graphs with diameter $2$ have unit acquisition number $1$ except for $C_5$ and the Petersen graph., Comment: 19 pages, 7 Figures

Let $\mathcal{F}$ be a family of graphs. A graph $G$ is $\mathcal{F}$-saturated if $G$ contains no member of $\mathcal{F}$ as a subgraph but $G+e$ contains some member of $\mathcal{F}$ whenever $e\in E(\overline{G})$. The saturation number and extremal number of $\mathcal{F}$, denoted $sat(n,\mathcal{F})$ and $ex(n,\mathcal{F})$ respectively, are the minimum and maximum numbers of edges among $n$-vertex $\mathcal{F}$-saturated graphs. For $k\in\mathbb{N}$, let $\mathcal{F}_k$ and $\mathcal{F}'_k$ be the families of $k$-connected and $k$-edge-connected graphs, respectively. Wenger proved $sat(n,\mathcal{F}_k)=(k-1)n-{k\choose2}$, we prove $sat(n,\mathcal{F}'_k)=(k-1)(n-1)-\lfloor{\frac {n}{k+1}}\rfloor{k-1 \choose 2}$. We also prove $ex(n,\mathcal{F}'_k)=(k-1)n-{k\choose2}$ and characterize when equality holds. Finally, we give a lower bound on the spectral radius for $\mathcal{F}_k$-saturated and $\mathcal{F}'_k$-saturated graphs., Comment: 9 pages, we welcome any comments and suggestions

In 2010, N. Lichiardopol conjectured for $q \geq 3$ and $k \geq 1$ that any tournament with minimum out-degree at least $(q-1)k-1$ contains $k$ disjoint cycles of length $q$. We prove this conjecture for $q \geq 5$. Since it is already known to hold for $q\le4$, this completes the proof of the conjecture.

The slow-coloring game is played by Lister and Painter on a graph $G$. On each round, Lister marks a nonempty subset $M$ of the remaining vertices, scoring $|M|$ points. Painter then gives a color to a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. Painter's goal is to minimize the total score; Lister seeks to maximize it. The score that each player can guarantee doing no worse than is the sum-color cost of $G$, written $\mathring{\rm s}(G)$. We develop a linear-time algorithm to compute $\mathring{\rm s}(G)$ when $G$ is a tree, enabling us to characterize the $n$-vertex trees with the largest and smallest values. Our algorithm also computes on trees the interactive sum choice number, a parameter recently introduced by Bonamy and Meeks., Comment: 18 pages, 2 figures. This version includes the proof that the sum-color cost agrees with the interactive sum choice number on trees (formerly a standalone paper)

The separation dimension of a graph $G$, written $\pi(G)$, is the minimum number of linear orderings of $V(G)$ such that every two nonincident edges are "separated" in some ordering, meaning that both endpoints of one edge appear before both endpoints of the other. We introduce the fractional separation dimension $\pi_f(G)$, which is the minimum of $a/b$ such that some $a$ linear orderings (repetition allowed) separate every two nonincident edges at least $b$ times. In contrast to separation dimension, fractional separation dimension is bounded: always $\pi_f(G)\le 3$, with equality if and only if $G$ contains $K_4$. There is no stronger bound even for bipartite graphs, since $\pi_f(K_{m,m})=\pi_f(K_{m+1,m})=\frac{3m}{m+1}$. We also compute $\pi_f(G)$ for cycles and some complete tripartite graphs. We show that $\pi_f(G)<\sqrt 2$ when $G$ is a tree and present a sequence of trees on which the value tends to $4/3$. Finally, we consider analogous problems for circular orderings, where pairs of nonincident edges are separated unless their endpoints alternate. Let $\pi^\circ(G)$ be the number of circular orderings needed to separate all pairs and $\pi_f^\circ(G)$ be the fractional version. Among our results: (1) $\pi^\circ(G)=1$ if and only $G$ is outerplanar. (2) $\pi^\circ(G)\le2$ when $G$ is bipartite. (3) $\pi^\circ(K_n)\ge\log_2\log_3(n-1)$. (4) $\pi_f^\circ(G)\le\frac{3}{2}$, with equality if and only if $K_4\subseteq G$. (5) $\pi_f^\circ(K_{m,m})=\frac{3m-3}{2m-1}$.

The $k$-deck of a graph is its multiset of induced subgraphs on $k$ vertices. We prove that $n$-vertex graphs with maximum degree $2$ have the same $k$-decks if each cycle has at least $k+1$ vertices, each path component has at least $k-1$ vertices, and the number of edges is the same. Using this for lower bounds, we obtain for each graph with maximum degree at most $2$ the least $k$ such that it is determined by its $k$-deck. For the $n$-vertex cycle this value is $\lfloor n/2 \rfloor$, and for the $n$-vertex path it is $\lfloor n/2 \rfloor+1$. Also, the least $k$ such that the $k$-deck of an $n$-vertex graph always determines whether it is connected is at least $\lfloor n/2 \rfloor +1$.

A spanning tree of a graph $G$ is a connected acyclic spanning subgraph of $G$. We consider enumeration of spanning trees when $G$ is a $2$-tree, meaning that $G$ is obtained from one edge by iteratively adding a vertex whose neighborhood consists of two adjacent vertices. We use this construction order both to inductively list the spanning trees without repetition and to give bounds on the number of them. We determine the $n$-vertex $2$-trees having the most and the fewest spanning trees. The $2$-tree with the fewest is unique; it has $n-2$ vertices of degree $2$ and has $n2^{n-3}$ spanning trees. Those with the most are all those having exactly two vertices of degree $2$, and their number of spanning trees is the Fibonacci number $F_{2n-2}$., Comment: 10 Pages, 4 Figures

For an interconnection network $G$, the {\it $\omega$-wide diameter} $d_\omega(G)$ is the least $\ell$ such that any two vertices are joined by $\omega$ internally-disjoint paths of length at most $\ell$, and the {\it $(\omega-1)$-fault diameter} $D_{\omega}(G)$ is the maximum diameter of a subgraph obtained by deleting fewer than $\omega$ vertices of $G$. The enhanced hypercube $Q_{n,k}$ is a variant of the well-known hypercube. Yang, Chang, Pai, and Chan gave an upper bound for $d_{n+1}(Q_{n,k})$ and $D_{n+1}(Q_{n,k})$ and posed the problem of finding the wide diameters and fault diameters of $Q_{n,k}$. By constructing internally disjoint paths between any two vertices in the enhanced hypercube, for $n\ge3$ and $2\le k\le n$ we prove $$ D_\omega(Q_{n,k})=d_\omega(Q_{n,k})=\begin{cases} d(Q_{n,k}) & \textrm{for $1 \leq \omega < n-\lfloor\frac{k}{2}\rfloor$;}\\ d(Q_{n,k})+1 & \textrm{for $n-\lfloor\frac{k}{2}\rfloor \leq \omega \leq n+1$.} \end{cases} $$ where $d(Q_{n,k})$ is the diameter of $Q_{n,k}$. These results mean that interconnection networks modelled by enhanced hypercubes are extremely robust., Comment: 9 pages, 1 figure

We give a new proof of the theorem of Boesch-Tindell and Farzad-Mahdian-Mahmoodian-Saberi-Sadri that a directed graph extends to a strongly connected digraph on the same vertex set if and only if it has no complete directed cut. Our proof bounds the number of edges needed for such an extension; we give examples to demonstrate sharpness. We apply the characterization to a problem on non-transitive dice., Comment: 8 pp. V2: 9 pp. Cite previous publication of one theorem. V3: Minor copyedits

The slow-coloring game is played by Lister and Painter on a graph $G$. On each round, Lister marks a nonempty subset $M$ of the uncolored vertices, scoring $|M|$ points. Painter then gives a color to a subset of $M$ that is independent in $G$. The game ends when all vertices are colored. Painter and Lister want to minimize and maximize the total score, respectively. The best score that each player can guarantee is the sum-color cost of $G$, written $\mathring{\mathrm{s}}(G)$. The game is an online variant of online sum list coloring. We proe $\frac{|V(G)|}{2\alpha(G)} + \frac{1}{2} \leq \frac{\mathring{\mathrm{s}}(G)}{|V(G)|} \leq \max\left\{ \frac{|V(H)|}{\alpha(H)} : H \subset G\right\}$, where $\alpha(G)$ is the independence number, and we study when equality holds in the bounds. We compute $\mathring{\mathrm{s}}(G)$ for graphs with $\alpha(G) = 2$. Among $n$-vertex graphs, we prove that $\mathring{\mathrm{s}}$ is minimized by the star and maximized by the path. We also obtain good bounds on $\mathring{\mathrm{s}}(K_{r,s})$., Comment: 18 pages. Revised introduction, restructured several proofs

Given a graph $F$, a graph $G$ is {\it uniquely $F$-saturated} if $F$ is not a subgraph of $G$ and adding any edge of the complement to $G$ completes exactly one copy of $F$. In this paper we study uniquely $C_t$-saturated graphs. We prove the following: (1) a graph is uniquely $C_5$-saturated if and only if it is a friendship graph. (2) There are no uniquely $C_6$-saturated graphs or uniquely $C_7$-saturated graphs. (3) For $t\ge6$, there are only finitely many uniquely $C_t$-saturated graphs (we conjecture that in fact there are none)., Comment: 14 pages, 5 figures

For a loopless multigraph $G$, the fractional arboricity $Arb(G)$ is the maximum of $\frac{|E(H)|}{|V(H)|-1}$ over all subgraphs $H$ with at least two vertices. Generalizing the Nash-Williams Arboricity Theorem, the Nine Dragon Tree Conjecture asserts that if $Arb(G)\le k+\frac{d}{k+d+1}$, then $G$ decomposes into $k+1$ forests with one having maximum degree at most $d$. The conjecture was previously proved for $d=k+1$ and for $k=1$ when $d \le 6$. We prove it for all $d$ when $k \le 2$, except for $(k,d)=(2,1)$., Comment: 15 pages

An $r$-augmented tree is a rooted tree plus $r$ edges added from each leaf to ancestors. For $d,g,r\in\mathbb{N}$, we construct a bipartite $r$-augmented complete $d$-ary tree having girth at least $g$. The height of such trees must grow extremely rapidly in terms of the girth. Using the resulting graphs, we construct sparse non-$k$-choosable bipartite graphs, showing that maximum average degree at most $2(k-1)$ is a sharp sufficient condition for $k$-choosability in bipartite graphs, even when requiring large girth. We also give a new simple construction of non-$k$-colorable graphs and hypergraphs with any girth $g$., Comment: Slight adjustments, including simplifying the proofs of Lemmas 2.2 and 2.3 and the arguments for large girth

An \textit{$r$-dynamic $k$-coloring} of a graph $G$ is a proper $k$-coloring of $G$ such that every vertex in $V(G)$ has neighbors in at least $\min\{d(v),r\}$ different color classes. The \textit{$r$-dynamic chromatic number} of a graph $G$, written $\chi_r(G)$, is the least $k$ such that $G$ has such a coloring. Proving a conjecture of Jahanbekam, Kim, O, and West, we show that the $m$-by-$n$ grid has no $3$-dynamic $4$-coloring when $mn\equiv2\mod 4$. This completes the determination of the $r$-dynamic chromatic number of the $m$-by-$n$ grid for all $r,m,n$.

We consider a Cops-and-Robber game played on the subsets of an $n$-set. The robber starts at the full set; the cops start at the empty set. On each turn, the robber moves down one level by discarding an element, and each cop moves up one level by gaining an element. The question is how many cops are needed to ensure catching the robber when the robber reaches the middle level. Aaron Hill posed the problem and provided a lower bound of $2^{n/2}$ for even $n$ and $\binom{n}{\lceil n/2 \rceil}2^{-\lfloor n/2 \rfloor}$ for odd $n$. We prove an upper bound (for all $n$) that is within a factor of $O(\ln n)$ times this lower bound., Comment: Minor revisions

Given a family ${\mathcal F}$ and a host graph $H$, a graph $G\subseteq H$ is ${\mathcal F}$-saturated relative to $H$ if no subgraph of $G$ lies in ${\mathcal F}$ but adding any edge from $E(H)-E(G)$ to $G$ creates such a subgraph. In the ${\mathcal F}$-saturation game on $H$, players Max and Min alternately add edges of $H$ to $G$, avoiding subgraphs in ${\mathcal F}$, until $G$ becomes ${\mathcal F}$-saturated relative to $H$. They aim to maximize or minimize the length of the game, respectively; $\textrm{sat}_g({\mathcal F};H)$ denotes the length under optimal play (when Max starts). Let ${\mathcal O}$ denote the family of all odd cycles and ${\mathcal T}$ the family of $n$-vertex trees, and write $F$ for ${\mathcal F}$ when ${\mathcal F}=\{F\}$. Our results include $\textrm{sat}_g({\mathcal O};K_{2k})=k^2$, $\textrm{sat}_g({\mathcal T};K_n)=\binom{n-2}{2}+1$ for $n\ge6$, $\textrm{sat}_g(K_{1,3};K_n)=2\lfloor n/2 \rfloor$ for $n\ge8$, $\textrm{sat}_g(K_{1,r+1};K_n)=\frac{rn}{2}-\frac{r^2}{8}+O(1)$, and $|\textrm{sat}_g(P_4;K_n)-(4n-1)/5|\le 1$. We also determine $\textrm{sat}_g(P_4;K_{m,n})$; with $m\ge n$, it is $n$ when $n$ is even, $m$ when $n$ is odd and $m$ is even, and $m+\lfloor n/2 \rfloor$ when $mn$ is odd. Finally, we prove the lower bound $\textrm{sat}_g(C_4;K_{n,n})\ge\frac{1}{10.4}n^{13/12}-O(n^{35/36})$. The results are very similar when Min plays first, except for the $P_4$-saturation game on $K_{m,n}$., Comment: updated with references to recent related work

Let $\text{ch}(G)$ denote the choice number of a graph $G$ (also called "list chromatic number" or "choosability" of $G$). Noel, Reed, and Wu proved the conjecture of Ohba that $\text{ch}(G)=\chi(G)$ when $|V(G)|\le 2\chi(G)+1$. We extend this to a general upper bound: $\text{ch}(G)\le \max\{\chi(G),\lceil({|V(G)|+\chi(G)-1})/{3}\rceil\}$. Our result is sharp for $|V(G)|\le 3\chi(G)$ using Ohba's examples, and it improves the best-known upper bound for $\text{ch}(K_{4,\dots,4})$., Comment: 14 pages

We provide a "how-to" guide to the use and application of the Discharging Method. Our aim is not to exhaustively survey results proved by this technique, but rather to demystify the technique and facilitate its wider use, using applications in graph coloring as examples. Along the way, we present some new proofs and new problems., Comment: We shortened the paper (now 50 pages, originally 78). This version omits total, entire, and improper coloring, as well as linear choosability. It spends less time on circular, acyclic, and star coloring, along with linear arboricity. (For those topics, see the first version.) We significantly reorganized the first few sections. We feel that the result is more accessible and inviting

We apply the Discharging Method to prove the 1,2,3-Conjecture and the 1,2-Conjecture for graphs with maximum average degree less than 8/3. Stronger results on these conjectures have been proved, but this is the first application of discharging to them, and the structure theorems and reducibility results are of independent interest., Comment: 29 pages, 19 figures; same results as earlier version, but revised introduction