arXiv subjects
Publications and source records attributed to Daniel W. Cranston.
We prove that if $G$ is a quasi-line graph with $Δ(G)>ω(G)$ and $Δ(G)\ge 69$, then $χ_{OL}(G)\le Δ(G)-1$. Together with our previous work, this implies that if $G$ is a claw-free graph with $Δ(G)>ω(G)$ and $Δ(G)\ge 69$, then $χ_{\ell}(G)\le Δ(G)-1$.
Brooks' Theorem states that if a graph has $Δ\ge 3$ and $ω\le Δ$, then $χ\le Δ$. Borodin and Kostochka conjectured that if $Δ\ge 9$ and $ω\le Δ-1$, then $χ\le Δ-1$. We show that if $Δ\ge 13$ and $ω\le Δ-4$, then $χ\le Δ-1$. For a graph $G$, let $\mathcal{H}$ denote the subgraph of $G$ induced by vertices of degree $Δ$. We also show that if $ω\le Δ-1$ and $ω(\mathcal{H})\le Δ-6$, then $χ\le Δ-1$.
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.
Let $G$ be a planar graph without 4-cycles and 5-cycles and with maximum degree $Δ\ge 32$. We prove that $χ_{\ell}(G^2)\le Δ+3$. For arbitrarily large maximum degree $Δ$, there exist planar graphs $G_Δ$ of girth 6 with $χ(G_Δ^2)=Δ+2$. 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, $λ_{2,1}(G)\le Δ+8$ and, more generally, $λ_{p,q}(G)\le (2q-1)Δ+6p-2q-2$, 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.
The {\em square} $G^2$ of a graph $G$ is the graph with the same vertex set as $G$ and with two vertices adjacent if their distance in $G$ is at most 2. Thomassen showed that every planar graph $G$ with maximum degree $Δ(G)=3$ satisfies $χ(G^2)\leq 7$. Kostochka and Woodall conjectured that for every graph, the list-chromatic number of $G^2$ equals the chromatic number of $G^2$, that is $χ_l(G^2)=χ(G^2)$ for all $G$. If true, this conjecture (together with Thomassen's result) implies that every planar graph $G$ with $Δ(G)=3$ satisfies $χ_l(G^2)\leq 7$. We prove that every connected graph (not necessarily planar) with $Δ(G)=3$ other than the Petersen graph satisfies $χ_l(G^2)\leq 8$ (and this is best possible). In addition, we show that if $G$ is a planar graph with $Δ(G)=3$ and girth $g(G)\geq 7$, then $χ_l(G^2)\leq 7$. Dvořák, Škrekovski, and Tancer showed that if $G$ is a planar graph with $Δ(G) = 3$ and girth $g(G) \geq 10$, then $χ_l(G^2)\leq 6$. We improve the girth bound to show that if $G$ is a planar graph with $Δ(G)=3$ and $g(G) \geq 9$, then $χ_l(G^2) \leq 6$. All of our proofs can be easily translated into linear-time coloring algorithms.
The chromatic number of the plane is the chromatic number of the uncountably infinite graph that has as its vertices the points of the plane and has an edge between two points if their distance is 1. This chromatic number is denoted $χ(\mathcal{R}^2)$. The problem was introduced in 1950, and shortly thereafter it was proved that $4\le χ(\mathcal{R}^2)\le 7$. These bounds are both easy to prove, but after more than 60 years they are still the best known. In this paper, we investigate $χ_f(\mathcal{R}^2)$, the fractional chromatic number of the plane. The previous best bounds (rounded to five decimal places) were $3.5556 \le χ_f(\mathcal{R}^2)\le 4.3599$. Here we improve the lower bound to $76/21\approx3.6190$.
We prove bounds on the chromatic number $χ$ of a vertex-transitive graph in terms of its clique number $ω$ and maximum degree $Δ$. We conjecture that every vertex-transitive graph satisfies $χ\le \max \left\{ω, \left\lceil\frac{5Δ+ 3}{6}\right\rceil\right\}$ and we prove results supporting this conjecture. Finally, for vertex-transitive graphs with $Δ\ge 13$ we prove the Borodin-Kostochka conjecture, i.e., $χ\le\max\{ω,Δ-1\}$.
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.
Cranston and Kim conjecture that if $G$ is a connected graph with maximum degree $Δ$ and $G$ is not a Moore Graph, then $χ_l(G^2) \le Δ^2-1$; here $χ_l$ is the list chromatic number. We prove their conjecture; in fact, this upper bound holds even for online list chromatic number.
We determine the list chromatic number of the square of a graph $\chil(G^2)$ in terms of its maximum degree $Δ$ when its maximum average degree, denoted $\mad(G)$, is sufficiently small. For $Δ\ge 6$, if $\mad(G)<2+\frac{4Δ-8}{5Δ+2}$, then $\chil(G^2)=Δ+1$. In particular, if $G$ is planar with girth $g\ge 7+\frac{12}{Δ-2}$, then $\chil(G^2)=Δ+1$. Under the same conditions, $\chil^i(G)=Δ$, where $\chil^i$ is the list injective chromatic number.
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 $Δ(G)$, then $T(G)$ is $(2Δ(G)-1)$-choosable. We give a linear-time algorithm that produces such a coloring. The best previous general upper bound for $Δ(G) > 3$ was $\floor{\frac32Δ(G)+2}$, by Borodin et al. When $Δ(G)=4$, our algorithm gives a better upper bound. When $Δ(G)\in\{3,5,6\}$, 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.).
A graph $G$ is called \emph{chromatic-choosable} if its choice number is equal to its chromatic number, namely $Ch(G)=χ(G)$. Ohba has conjectured that every graph $G$ satisfying $|V(G)|\leq 2χ(G)+1$ 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).
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.
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.
We study squares of planar graphs with the aim to determine their list chromatic number. We present new upper bounds for the square of a planar graph with maximum degree $Δ\leq 4$. In particular $G^2$ is 5-, 6-, 7-, 8-, 12-, 14-choosable if the girth of $G$ is at least 16, 11, 9, 7, 5, 3 respectively. In fact we prove more general results, in terms of maximum average degree, that imply the results above.
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.
We study a competitive optimization version of $α'(G)$, 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 $\Max(G)=α'(G)$ that is preserved under cartesian product. In general, $\Max(G)\ge \frac23α'(G)$, with equality for many split graphs, while $\Max(G)\ge\frac34α'(G)$ 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 $n/7$.
We prove that every claw-free graph $G$ that doesn't contain a clique on $Δ(G) \geq 9$ vertices can be $Δ(G) - 1$ colored.