Linear and 2-Frugal Choosability of Graphs of Small Maximum Average Degree

International audienceA proper vertex colouring of a graph G is 2-frugal (resp. linear) if the graph induced by the vertices of any two colour classes is of maximum degree 2 (resp. is a forest of paths). A graph G is 2-frugally (resp. linearly) L-colourable if for a given list assignment L : V(G) → N, there exists a 2-frugal (resp. linear) colouring c of G such that c(v) ∈ L(v) for all v ∈ V (G). If G is 2-frugally (resp. linearly) L-list colourable for any list assignment such that |L(v)| ≥ k for all v ∈ V (G), then G is 2-frugally (resp. linearly) k-choosable. In this paper, we improve some bounds on the 2-frugal choosability and linear choosability of graphs with small maximum average degree

k-forested choosability of graphs with bounded maximum average degree

A proper vertex coloring of a simple graph is $k$-forested if the graph induced by the vertices of any two color classes is a forest with maximum degree less than $k$. A graph is $k$-forested $q$-choosable if for a given list of $q$ colors associated with each vertex $v$, there exists a $k$-forested coloring of $G$ such that each vertex receives a color from its own list. In this paper, we prove that the $k$-forested choosability of a graph with maximum degree $\Delta\geq k\geq 4$ is at most $\lceil\frac{\Delta}{k-1}\rceil+1$, $\lceil\frac{\Delta}{k-1}\rceil+2$ or $\lceil\frac{\Delta}{k-1}\rceil+3$ if its maximum average degree is less than 12/5, $8/3 or 3, respectively.Comment: Please cite this paper in press as X. Zhang, G. Liu, J.-L. Wu, k-forested choosability of graphs with bounded maximum average degree, Bulletin of the Iranian Mathematical Society, to appea

Improved bounds on coloring of graphs

Given a graph $G$ with maximum degree $\Delta\ge 3$, we prove that the acyclic edge chromatic number $a'(G)$ of $G$ is such that $a'(G)\le\lceil 9.62 (\Delta-1)\rceil$. Moreover we prove that: $a'(G)\le \lceil 6.42(\Delta-1)\rceil$ if $G$ has girth $g\ge 5\,$; a'(G)\le \lceil5.77 (\Delta-1)\rc if $G$ has girth $g\ge 7$; a'(G)\le \lc4.52(\D-1)\rc if $g\ge 53$; a'(G)\le \D+2\, if g\ge \lceil25.84\D\log\D(1+ 4.1/\log\D)\rceil. We further prove that the acyclic (vertex) chromatic number $a(G)$ of $G$ is such that a(G)\le \lc 6.59 \Delta^{4/3}+3.3\D\rc. We also prove that the star-chromatic number $\chi_s(G)$ of $G$ is such that \chi_s(G)\le \lc4.34\Delta^{3/2}+ 1.5\D\rc. We finally prove that the \b-frugal chromatic number \chi^\b(G) of $G$ is such that \chi^\b(G)\le \lc\max\{k_1(\b)\D,\; k_2(\b){\D^{1+1/\b}/ (\b!)^{1/\b}}\}\rc, where k_1(\b) and k_2(\b) are decreasing functions of \b such that k_1(\b)\in[4, 6] and k_2(\b)\in[2,5]. To obtain these results we use an improved version of the Lov\'asz Local Lemma due to Bissacot, Fern\'andez, Procacci and Scoppola \cite{BFPS}.Comment: Introduction revised. Added references. Corrected typos. Proof of Theorem 2 (items c-f) written in more detail

Acyclic and frugal colourings of graphs

Given a graph G = (V, E), a proper vertex colouring of V is t-frugal if no colour appears more than t times in any neighbourhood and is acyclic if each of the bipartite graphs consisting of the edges between any two colour classes is acyclic. For graphs of bounded maximum degree, Hind, Molloy and Reed [14] studied proper t-frugal colourings and Yuster [19] studied acyclic proper 2-frugal colourings. In this paper, we expand and generalise this study. In particular, we consider vertex colourings that are not necessarily proper, and in this case, we find qualitative connections with colourings that are t-improper -colourings in which the colour classes induce subgraphs of maximum degree at most t -for choices of t near to d

Conflict-Free Colourings of Graphs and Hypergraphs

A colouring of the vertices of a hypergraph H is called conflict-free if each hyperedge E of H contains a vertex of ‘unique' colour that does not get repeated in E. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of H, and is denoted by χCF(H). This parameter was first introduced by Even, Lotker, Ron and Smorodinsky (FOCS 2002) in a geometric setting, in connection with frequency assignment problems for cellular networks. Here we analyse this notion for general hypergraphs. It is shown that $\chi_{\rm CF}(H)\leq 1/2+\sqrt{2m+1/4}$, for every hypergraph with m edges, and that this bound is tight. Better bounds of the order of m1/t log m are proved under the assumption that the size of every edge of H is at least 2t − 1, for some t ≥ 3. Using Lovász's Local Lemma, the same result holds for hypergraphs in which the size of every edge is at least 2t − 1 and every edge intersects at most m others. We give efficient polynomial-time algorithms to obtain such colourings. Our machinery can also be applied to the hypergraphs induced by the neighbourhoods of the vertices of a graph. It turns out that in this case we need far fewer colours. For example, it is shown that the vertices of any graph G with maximum degree Δ can be coloured with log2+ε Δ colours, so that the neighbourhood of every vertex contains a point of ‘unique' colour. We give an efficient deterministic algorithm to find such a colouring, based on a randomized algorithmic version of the Lovász Local Lemma, suggested by Beck, Molloy and Reed. To achieve this, we need to (1) correct a small error in the Molloy-Reed approach, (2) restate and re-prove their result in a deterministic for

Linear Choosability of Sparse Graphs

We study the linear list chromatic number, denoted \lcl(G), of sparse graphs. The maximum average degree of a graph $G$, denoted \mad(G), is the maximum of the average degrees of all subgraphs of $G$. It is clear that any graph $G$ with maximum degree $\Delta(G)$ satisfies \lcl(G)\ge \ceil{\Delta(G)/2}+1. In this paper, we prove the following results: (1) if \mad(G)<12/5 and $\Delta(G)\ge 3$, then \lcl(G)=\ceil{\Delta(G)/2}+1, and we give an infinite family of examples to show that this result is best possible; (2) if \mad(G)<3 and $\Delta(G)\ge 9$, then \lcl(G)\le\ceil{\Delta(G)/2}+2, and we give an infinite family of examples to show that the bound on \mad(G) cannot be increased in general; (3) if $G$ is planar and has girth at least 5, then \lcl(G)\le\ceil{\Delta(G)/2}+4.Comment: 12 pages, 2 figure