Defective and Clustered Graph Colouring

Consider the following two ways to colour the vertices of a graph where the requirement that adjacent vertices get distinct colours is relaxed. A colouring has "defect" $d$ if each monochromatic component has maximum degree at most $d$. A colouring has "clustering" $c$ if each monochromatic component has at most $c$ vertices. This paper surveys research on these types of colourings, where the first priority is to minimise the number of colours, with small defect or small clustering as a secondary goal. List colouring variants are also considered. The following graph classes are studied: outerplanar graphs, planar graphs, graphs embeddable in surfaces, graphs with given maximum degree, graphs with given maximum average degree, graphs excluding a given subgraph, graphs with linear crossing number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdi\`ere parameter, graphs with given circumference, graphs excluding a fixed graph as an immersion, graphs with given thickness, graphs with given stack- or queue-number, graphs excluding $K_t$ as a minor, graphs excluding $K_{s,t}$ as a minor, and graphs excluding an arbitrary graph $H$ as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in the Electronic Journal of Combinatoric

Coloring and covering problems on graphs

The \emph{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 \emph{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, we show 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 $4/3$. We conjecture that when $n=3m$ the $K_4$-free $n$-vertex graph maximizing $\pi_f(G)$ is $K_{m,m,m}$. We also 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 let $\pi_f^\circ(G)$ 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) $\pi_f^\circ(K_{m,m})=\frac{3m-3}{2m-1}$. A \emph{star $k$-coloring} is a proper $k$-coloring where the union of any two color classes induces a star forest. While every planar graph is 4-colorable, not every planar graph is star 4-colorable. One method to produce a star 4-coloring is to partition the vertex set into a 2-independent set and a forest; such a partition is called an \emph{\Ifp}. We use discharging to prove that every graph with maximum average degree less than $\frac{5}{2}$ has an \Ifp, which is sharp and improves the result of Bu, Cranston, Montassier, Raspaud, and Wang (2009). As a corollary, we gain that every planar graph with girth at least 10 has a star 4-coloring. A proper vertex coloring of a graph $G$ is \emph{$r$-dynamic} if for each $v\in V(G)$, at least $\min\{r,d(v)\}$ colors appear in $N_G(v)$. We investigate $3$-dynamic versions of coloring and list coloring. We prove that planar and toroidal graphs are 3-dynamically 10-choosable, and this bound is sharp for toroidal graphs. Given a proper total $k$-coloring $c$ of a graph $G$, we define the \emph{sum value} of a vertex $v$ to be $c(v) + \sum_{uv \in E(G)} c(uv)$. The smallest integer $k$ such that $G$ has a proper total $k$-coloring whose sum values form a proper coloring is the \emph{neighbor sum distinguishing total chromatic number} $\chi''_{\Sigma}(G)$. Pil{\'s}niak and Wo{\'z}niak~(2013) conjectured that $\chi''_{\Sigma}(G)\leq \Delta(G)+3$ for any simple graph with maximum degree $\Delta(G)$. We prove this bound to be asymptotically correct by showing that $\chi''_{\Sigma}(G)\leq \Delta(G)(1+o(1))$. The main idea of our argument relies on Przyby{\l}o's proof (2014) for neighbor sum distinguishing edge-coloring

Proceedings of the 8th Cologne-Twente Workshop on Graphs and Combinatorial Optimization

International audienceThe Cologne-Twente Workshop (CTW) on Graphs and Combinatorial Optimization started off as a series of workshops organized bi-annually by either Köln University or Twente University. As its importance grew over time, it re-centered its geographical focus by including northern Italy (CTW04 in Menaggio, on the lake Como and CTW08 in Gargnano, on the Garda lake). This year, CTW (in its eighth edition) will be staged in France for the first time: more precisely in the heart of Paris, at the Conservatoire National d’Arts et Métiers (CNAM), between 2nd and 4th June 2009, by a mixed organizing committee with members from LIX, Ecole Polytechnique and CEDRIC, CNAM