Search

Hocquard, Kim, and Pierron constructed, for every even integer $D\ge 2$, a 2-degenerate graph $G_D$ with maximum degree $D$ such that $\omega(G_D^2)=\frac52D$. They asked whether (a) there exists $D_0$ such that every 2-degenerate graph $G$ with maximum degree $D\ge D_0$ satisfies $\chi(G^2)\le \frac52D$ and (b) whether this result holds more generally for every graph $G$ with mad(G), Comment: 15 pages, 7 figures

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

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 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 $\Delta(G)\ge 10^{9}$; 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., 32 pages, 13 figures and tables, plus 195-entry bibliography, comments are welcome, published as a Dynamic Survey in Electronic Journal of Combinatorics

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., 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 also prove more precise results when $G$ is a tree. 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: 18 pages, 3 figures

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., 8 pages

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 planar 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

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), Comment: 11 pages, 5 figures; to appear in European J. Combinatorics

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

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: 27 pages, 8 figures; second version extends the main result to cliques, which were previously excluded

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

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., 21 pages, 9 figures and 2 tables; this version incorporates reviewer feedback; to appear in SIAM J. Discrete Math

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

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.

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., 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

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}, $��'_a(G)$, is the smallest number of colors allowing an acyclic edge-coloring of $G$. Clearly $��'_a(G)\ge ��(G)$ for every graph $G$. Cohen, Havet, and M��ller conjectured that there exists a constant $M$ such that every planar graph with $��(G)\ge M$ has $��'_a(G)=��(G)$. We prove this conjecture., 15 pages, 8 figures; minor revisions to incorporate reviewer feedback; to appear in SIAM J. Discrete Math

Many graph coloring proofs proceed by showing that a minimal counterexample to the theorem being proved cannot contain certain configurations, and then showing that each graph under consideration contains at least one such configuration; these configurations are called \emph{reducible} for that theorem. (A \emph{configuration} is a subgraph $H$, along with specified degrees $d_G(v)$ in the original graph $G$ for each vertex of $H$.) We give a general framework for showing that configurations are reducible for edge-coloring. A particular form of reducibility, called \emph{fixability}, can be considered without reference to a containing graph. This has two key benefits: (i) we can now formulate necessary conditions for fixability, and (ii) the problem of fixability is easy for a computer to solve. The necessary condition of \emph{superabundance} is sufficient for multistars and we conjecture that it is sufficient for trees as well, which would generalize the powerful technique of Tashkinov trees. Via computer, we can generate thousands of reducible configurations, but we have short proofs for only a small fraction of these. The computer can write \LaTeX\ code for its proofs, but they are only marginally enlightening and can run thousands of pages long. We give examples of how to use some of these reducible configurations to prove conjectures on edge-coloring for small maximum degree. Our aims in writing this paper are (i) to provide a common context for a variety of reducible configurations for edge-coloring and (ii) to spur development of methods for humans to understand what the computer already knows., 18 pages, 8 figures; 12-page appendix with 39 figures

We consider graphs $G$ with $\Delta=3$ such that $\chi'(G)=4$ and $\chi'(G-e)=3$ for every edge $e$, so-called \emph{critical} graphs. Jakobsen noted that the Petersen graph with a vertex deleted, $P^*$, is such a graph and has average degree only $\frac83$. He showed that every critical graph has average degree at least $\frac83$, and asked if $P^*$ is the only graph where equality holds. A result of Cariolaro and Cariolaro shows that this is true. We strengthen this average degree bound further. Our main result is that if $G$ is a subcubic critical graph other than $P^*$, then $G$ has average degree at least $\frac{46}{17}\approx2.706$. This bound is best possible, as shown by the Hajos join of two copies of $P^*$., Comment: 16 pages, 10 figures; version 2 incorporates referee feedback, which led to improved exposition and a slightly stronger bound

Let $G$ be a planar graph without 4-cycles and 5-cycles and with maximum degree $\Delta\ge 32$. We prove that $\chi_{\ell}(G^2)\le \Delta+3$. For arbitrarily large maximum degree $\Delta$, there exist planar graphs $G_{\Delta}$ of girth 6 with $\chi(G_{\Delta}^2)=\Delta+2$. Thus, our bound is within 1 of being optimal. Further, our bound comes from coloring greedily in a good order, so the bound immediately extends to online list-coloring. In addition, we prove bounds for $L(p,q)$-labeling. Specifically, $\lambda_{2,1}(G)\le \Delta+8$ and, more generally, $\lambda_{p,q}(G)\le (2q-1)\Delta+6p-2q-2$, for positive integers $p$ and $q$ with $p\ge q$. Again, these bounds come from a greedy coloring, so they immediately extend to the list-coloring and online list-coloring variants of this problem., Comment: 15 pages, 12 figures

We prove a conjecture of Dvo����k, Kr��l, Nejedl��, and ��krekovski that planar graphs of girth at least five are square $(��+2)$-colorable for large enough $��$. In fact, we prove the stronger statement that such graphs are square $(��+2)$-choosable and even square $(��+2)$-paintable., 22 pages, 7 figures; this version incorporates reviewer feedback: fixed and rewrote proof of Lemma 3.5, expanded Section 4; to appear in J. Combin. Theory Ser. B

We prove that if $G$ is a quasi-line graph with $��(G)>��(G)$ and $��(G)\ge 69$, then $��_{OL}(G)\le ��(G)-1$. Together with our previous work, this implies that if $G$ is a claw-free graph with $��(G)>��(G)$ and $��(G)\ge 69$, then $��_{\ell}(G)\le ��(G)-1$., 24 pages, 5 figures

We collect some of our favorite proofs of Brooks' Theorem, highlighting advantages and extensions of each. The proofs illustrate some of the major techniques in graph coloring, such as greedy coloring, Kempe chains, hitting sets, and the Kernel Lemma. We also discuss standard strengthenings of vertex coloring, such as list coloring, online list coloring, and Alon--Tarsi orientations, since analogues of Brooks' Theorem hold in each context. We conclude with two conjectures along the lines of Brooks' Theorem that are much stronger, the Borodin--Kostochka Conjecture and Reed's Conjecture., Survey paper of Brooks' Theorem and its extensions. 25 pages, 12 figures

A graph $G$ is called \emph{chromatic-choosable} if its choice number is equal to its chromatic number, namely $Ch(G)=\chi(G)$. Ohba has conjectured that every graph $G$ satisfying $|V(G)|\leq 2\chi(G)+1$ is chromatic-choosable. Since each $k$-chromatic graph is a subgraph of a complete $k$-partite graph, we see that Ohba's conjecture is true if and only if it is true for every complete multipartite graph. However, the only complete multipartite graphs for which Ohba's conjecture has been verified are: $K_{3*2,2*(k-3),1}$, $K_{3,2*(k-1)}$, $K_{s+3,2*(k-s-1),1*s}$, $K_{4,3,2*(k-4),1*2}$, and $K_{5,3,2*(k-5),1*3}$. In this paper, we show that Ohba's conjecture is true for two new classes of complete multipartite graphs: graphs with three parts of size 3 and graphs with one part of size 4 and two parts of size 3. Namely, we prove that $Ch(K_{3*3,2*(k-5),1*2})=k$ and $Ch(K_{4,3*2,2*(k-6),1*3})=k$ (for $k\geq 5$ and $k\geq 6$, respectively)., Comment: 10 pages; this paper proves 2 special cases of Ohba's Conjecture, which has now been proved completely: http://arxiv.org/abs/1211.1999

We investigate the boundedness of positive solutions of the reciprocal max-type difference equation \[ x_{n}=\max\left\{\frac{A_{n-1}^{1}}{x_{n-1}}, \frac{A_{n-1}^{2}}{x_{n-2}}, \ldots, \frac{A_{n-1}^{t}}{x_{n-t}}\right\}, \ \ n=1, 2, \ldots, \] where, for each value of $i$, the sequence $\{A_{n}^{i}\}_{n=0}^{\infty}$ of positive numbers is periodic with period $p_{i}$. We give both sufficient conditions on the $p_{i}$'s for the boundedness of all solutions and sufficient conditions for all solutions to be unbounded. This work essentially complements the work by Biddell and Franke, who showed that as long as every positive solution of our equation is \emph{bounded}, then every positive solution is eventually periodic, thereby leaving open the question as to when solutions are bounded., 18 pages

An antimagic labeling of a graph $G$ with $m$ edges is a bijection from $E(G)$ to $\{1,2,\ldots,m\}$ such that for all vertices $u$ and $v$, the sum of labels on edges incident to $u$ differs from that for edges incident to $v$. Hartsfield and Ringel conjectured that every connected graph other than the single edge $K_2$ has an antimagic labeling. We prove this conjecture for regular graphs of odd degree., 5 pages

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

Cranston and Kim conjecture that if $G$ is a connected graph with maximum degree $��$ and $G$ is not a Moore Graph, then $��_l(G^2) \le ��^2-1$; here $��_l$ is the list chromatic number. We prove their conjecture; in fact, this upper bound holds even for online list chromatic number., 23 pages, 10 figures. In version 2, we add a conjecture and 2 figures to the introduction. Results are unchanged

The \emph{total graph} $T(G)$ of a multigraph $G$ has as its vertices the set of edges and vertices of $G$ and has an edge between two vertices if their corresponding elements are either adjacent or incident in $G$. We show that if $G$ has maximum degree $��(G)$, then $T(G)$ is $(2��(G)-1)$-choosable. We give a linear-time algorithm that produces such a coloring. The best previous general upper bound for $��(G) > 3$ was $\floor{\frac32��(G)+2}$, by Borodin et al. When $��(G)=4$, our algorithm gives a better upper bound. When $��(G)\in\{3,5,6\}$, our algorithm matches the best known bound. However, because our algorithm is significantly simpler, it runs in linear time (unlike the algorithm of Borodin et al.)., 6 pages, 2 figures

We determine the list chromatic number of the square of a graph $\chil(G^2)$ in terms of its maximum degree $��$ when its maximum average degree, denoted $\mad(G)$, is sufficiently small. For $��\ge 6$, if $\mad(G), 16 pages, 6 figures; incorporated reviewer feedback

Brooks' Theorem states that if a graph has $��\ge 3$ and $��\le ��$, then $��\le ��$. Borodin and Kostochka conjectured that if $��\ge 9$ and $��\le ��-1$, then $��\le ��-1$. We show that if $��\ge 13$ and $��\le ��-4$, then $��\le ��-1$. For a graph $G$, let $\mathcal{H}$ denote the subgraph of $G$ induced by vertices of degree $��$. We also show that if $��\le ��-1$ and $��(\mathcal{H})\le ��-6$, then $��\le ��-1$., 27 pages, 3 figures; added many more details in this version, as well as a discussion of algorithms; to appear in SIAM J. Discrete Math

We study a competitive optimization version of $\alpha'(G)$, the maximum size of a matching in a graph $G$. Players alternate adding edges of $G$ to a matching until it becomes a maximal matching. One player (Max) wants that matching to be large; the other (Min) wants it to be small. The resulting sizes under optimal play when Max or Min starts are denoted $\Max(G)$ and $\Min(G)$, respectively. We show that always $|\Max(G)-\Min(G)|\le 1$. We obtain a sufficient condition for $\Max(G)=\alpha'(G)$ that is preserved under cartesian product. In general, $\Max(G)\ge \frac23\alpha'(G)$, with equality for many split graphs, while $\Max(G)\ge\frac34\alpha'(G)$ when $G$ is a forest. Whenever $G$ is a 3-regular $n$-vertex connected graph, $\Max(G) \ge n/3$, and there are such examples with $\Max(G)\le 7n/18$. For an $n$-vertex path or cycle, the answer is roughly $n/7$., Comment: 16 pages; this version improves explanations at a number of points and adds a few more examples

Borodin and Kostochka conjectured that every graph $G$ with maximum degree $��\ge 9$ satisfies $��\le \max\{��, ��-1\}$. We carry out an in-depth study of minimum counterexamples to the Borodin-Kostochka conjecture. Our main tool is the identification of graph joins that are $f$-choosable, where $f(v) = d(v) - 1$ for each vertex $v$. Since such a join cannot be an induced subgraph of a vertex critical graph with $��= ��$, we have a wealth of structural information about minimum counterexamples to the Borodin-Kostochka conjecture. Our main result proves that certain conjectures that are prima facie weaker than the Borodin-Kostochka conjecture are in fact equivalent to it. One such equivalent conjecture is the following: Any graph with $��\ge ��= 9$ contains $K_3 * \bar{K_6}$ as a subgraph., compact version only proving the list coloring results needed for the conjecture equivalences; 26 pages, 6 figures; final version accepted for publication

We prove that every claw-free graph $G$ that doesn't contain a clique on $��(G) \geq 9$ vertices can be $��(G) - 1$ colored., 18 pages, 4 figures; moved list-coloring section earlier; added overview at end of Section 1 and start of Section 2

A {\it list assignment} $L$ of a graph $G$ is a function that assigns a set (list) $L(v)$ of colors to every vertex $v$ of $G$. Graph $G$ is called {\it $L$-list colorable} if it admits a vertex coloring $\phi$ such that $\phi(v)\in L(v)$ for all $v\in V(G)$ and $\phi(v)\not=\phi(w)$ for all $vw\in E(G)$. The following question was raised by Bruce Richter. Let $G$ be a planar, 3-connected graph that is not a complete graph. Denoting by $d(v)$ the degree of vertex $v$, is $G$ $L$-list colorable for every list assignment $L$ with $|L(v)|=\min \{d(v), 6\}$ for all $v\in V(G)$? More generally, we ask for which pairs $(r,k)$ the following question has an affirmative answer. Let $r$ and $k$ be integers and let $G$ be a $K_5$-minor-free $r$-connected graph that is not a Gallai tree (i.e., at least one block of $G$ is neither a complete graph nor an odd cycle). Is $G$ $L$-list colorable for every list assignment $L$ with $|L(v)|=\min\{d(v),k\}$ for all $v\in V(G)$? We investigate this question by considering the components of $G[S_k]$, where $S_k:=\{v\in V(G) | d(v), Comment: 14 pages, 2 figures, 2 tables

An {\it overlap representation} of a graph $G$ assigns sets to vertices so that vertices are adjacent if and only if their assigned sets intersect with neither containing the other. The {\it overlap number} $\ol(G)$ (introduced by Rosgen) is the minimum size of the union of the sets in such a representation. We prove the following: (1) An optimal overlap representation of a tree can be produced in linear time, and its size is the number of vertices in the largest subtree in which the neighbor of any leaf has degree 2. (2) If $\delta(G)\ge 2$ and $G\ne K_3$, then $\ol(G)\le |E(G)|-1$, with equality when $G$ is connected and triangle-free and has no star-cutset. (3) If $G$ is an $n$-vertex plane graph with $n\ge5$, then $\ol(G)\le 2n-5$, with equality when every face has length 4 and there is no star-cutset. (4) If $G$ is an $n$-vertex graph with $n\ge 14$, then $\ol(G)\le \floor{n^2/4-n/2-1}$, and this is sharp (for even $n$, equality holds when $G$ arises from $K_{n/2,n/2}$ by deleting a perfect matching)., Comment: 21 pages

A labeling of a graph is a bijection from $E(G)$ to the set $\{1, 2,..., |E(G)|\}$. A labeling is \textit{antimagic} if for any distinct vertices $u$ and $v$, the sum of the labels on edges incident to $u$ is different from the sum of the labels on edges incident to $v$. We say a graph is antimagic if it has an antimagic labeling. In 1990, Ringel conjectured that every connected graph other than $K_2$ is antimagic. In this paper, we show that every regular bipartite graph (with degree at least 2) is antimagic. Our technique relies heavily on the Marriage Theorem., Comment: 9 pages