The square of a planar cubic graph is 77-colorable

We prove the conjecture made by G.Wegner in 1977 that the square of every planar, cubic graph is $7$-colorable. Here, $7$ cannot be replaced by $6$

Computing Square Colorings on Bounded-Treewidth and Planar Graphs

A square coloring of a graph $G$ is a coloring of the square $G^2$ of $G$, that is, a coloring of the vertices of $G$ such that any two vertices that are at distance at most $2$ in $G$ receive different colors. We investigate the complexity of finding a square coloring with a given number of $q$ colors. We show that the problem is polynomial-time solvable on graphs of bounded treewidth by presenting an algorithm with running time $n^{2^{\operatorname{tw} + 4}+O(1)}$ for graphs of treewidth at most $\operatorname{tw}$. The somewhat unusual exponent $2^{\operatorname{tw}}$ in the running time is essentially optimal: we show that for any $\epsilon>0$, there is no algorithm with running time $f(\operatorname{tw})n^{(2-\epsilon)^{\operatorname{tw}}}$ unless the Exponential-Time Hypothesis (ETH) fails. We also show that the square coloring problem is NP-hard on planar graphs for any fixed number $q \ge 4$ of colors. Our main algorithmic result is showing that the problem (when the number of colors $q$ is part of the input) can be solved in subexponential time $2^{O(n^{2/3}\log n)}$ on planar graphs. The result follows from the combination of two algorithms. If the number $q$ of colors is small ($\le n^{1/3}$), then we can exploit a treewidth bound on the square of the graph to solve the problem in time $2^{O(\sqrt{qn}\log n)}$. If the number of colors is large ($\ge n^{1/3}$), then an algorithm based on protrusion decompositions and building on our result for the bounded-treewidth case solves the problem in time $2^{O(n\log n/q)}$.Comment: 72 pages, 15 figures, full version of a paper accepted at SODA 202

A Study on Graph Coloring and Digraph Connectivity

This dissertation focuses on coloring problems in graphs and connectivity problems in digraphs. We obtain the following advances in both directions.;1. Results in graph coloring. For integers k,r \u3e 0, a (k,r)-coloring of a graph G is a proper coloring on the vertices of G with k colors such that every vertex v of degree d( v) is adjacent to vertices with at least min{lcub}d( v),r{rcub} different colors. The r-hued chromatic number, denoted by chir(G ), is the smallest integer k for which a graph G has a (k,r)-coloring.;For a k-list assignment L to vertices of a graph G, a linear (L,r)-coloring of a graph G is a coloring c of the vertices of G such that for every vertex v of degree d(v), c(v)∈ L(v), the number of colors used by the neighbors of v is at least min{lcub}dG(v), r{rcub}, and such that for any two distinct colors i and j, every component of G[c --1({lcub}i,j{rcub})] must be a path. The linear list r-hued chromatic number of a graph G, denoted chiℓ L,r(G), is the smallest integer k such that for every k-list L, G has a linear (L,r)-coloring. Let Mad( G) denotes the maximum subgraph average degree of a graph G. We prove the following. (i) If G is a K3,3-minor free graph, then chi2(G) ≤ 5 and chi3(G) ≤ 10. Moreover, the bound of chi2( G) ≤ 5 is best possible. (ii) If G is a P4-free graph, then chir(G) ≤q chi( G) + 2(r -- 1), and this bound is best possible. (iii) If G is a P5-free bipartite graph, then chir( G) ≤ rchi(G), and this bound is best possible. (iv) If G is a P5-free graph, then chi2(G) ≤ 2chi(G), and this bound is best possible. (v) If G is a graph with maximum degree Delta, then each of the following holds. (i) If Delta ≥ 9 and Mad(G) \u3c 7/3, then chiℓL,r( G) ≤ max{lcub}lceil Delta/2 rceil + 1, r + 1{rcub}. (ii) If Delta ≥ 7 and Mad(G)\u3c 12/5, then chiℓ L,r(G)≤ max{lcub}lceil Delta/2 rceil + 2, r + 2{rcub}. (iii) If Delta ≥ 7 and Mad(G) \u3c 5/2, then chi ℓL,r(G)≤ max{lcub}lcei Delta/2 rceil + 3, r + 3{rcub}. (vi) If G is a K 4-minor free graph, then chiℓL,r( G) ≤ max{lcub}r,lceilDelta/2\rceil{rcub} + lceilDelta/2rceil + 2. (vii) Every planar graph G with maximum degree Delta has chiℓL,r(G) ≤ Delta + 7.;2. Results in digraph connectivity. For a graph G, let kappa( G), kappa\u27(G), delta(G) and tau( G) denote the connectivity, the edge-connectivity, the minimum degree and the number of edge-disjoint spanning trees of G, respectively. Let f(G) denote kappa(G), kappa\u27( G), or Delta(G), and define f¯( G) = max{lcub}f(H): H is a subgraph of G{rcub}. An edge cut X of a graph G is restricted if X does not contain all edges incident with a vertex in G. The restricted edge-connectivity of G, denoted by lambda2(G), is the minimum size of a restricted edge-cut of G. We define lambda 2(G) = max{lcub}lambda2(H): H ⊂ G{rcub}.;For a digraph D, let kappa;(D), lambda( D), delta--(D), and delta +(D) denote the strong connectivity, arc-strong connectivity, minimum in-degree, and out-degree of D, respectively. For each f ∈ {lcub}kappa,lambda, delta--, +{rcub}, define f¯(D) = max{lcub} f(H): H is a subdigraph of D{rcub}.;Catlin et al. in [Discrete Math., 309 (2009), 1033-1040] proved a characterization of kappa\u27(G) in terms of tau(G). We proved a digraph version of this characterization by showing that a digraph D is k-arc-strong if and only if for any vertex v in D, D has k-arc-disjoint spanning arborescences rooted at v. We also prove a characterization of uniformly dense digraphs analogous to the characterization of uniformly dense undirected graphs in [Discrete Applied Math., 40 (1992) 285--302]. (Abstract shortened by ProQuest.)

Computing Square Colorings on Bounded-Treewidth and Planar Graphs

A {\em square coloring} of a graph $G$ is a coloring of the square $G^2$ of $G$, that is, a coloring of the vertices of $G$ such that any two vertices that are at distance at most $2$ in $G$ receive different colors. We investigate the complexity of finding a square coloring with a given number of $q$ colors. We show that the problem is polynomial-time solvable on graphs of bounded treewidth by presenting an algorithm with running time n^{2^{\ttw + 4}+O(1)} for graphs of treewidth at most \ttw. The somewhat unusual exponent 2^\ttw in the running time is essentially optimal: we show that for any $\epsilon>0$, there is no algorithm with running time f(\ttw)n^{(2-\epsilon)^\ttw} unless the Exponential-Time Hypothesis (ETH) fails. We also show that the square coloring problem is NP-hard on planar graphs for any fixed number $q \ge 4$ of colors. Our main algorithmic result is showing that the problem (when the number of colors $q$ is part of the input) can be solved in subexponential time $2^{O(n^{2/3}\log n)}$ on planar graphs. The result follows from the combination of two algorithms. If the number $q$ of colors is small ($\le n^{1/3}$), then we can exploit a treewidth bound on the square of the graph to solve the problem in time $2^{O(\sqrt{qn}\log n)}$. If the number of colors is large ($\ge n^{1/3}$), then an algorithm based on protrusion decompositions and building on our result for the bounded-treewidth case solves the problem in time $2^{O(n\log n/q)}$