214 research outputs found
Decomposition of bounded degree graphs into -free subgraphs
We prove that every graph with maximum degree admits a partition of
its edges into parts (as ) none of which
contains as a subgraph. This bound is sharp up to a constant factor. Our
proof uses an iterated random colouring procedure.Comment: 8 pages; to appear in European Journal of Combinatoric
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) â N, there exists a 2-frugal (resp. linear) colouring c of G such that c(v) â L(v) for all v â V (G). If G is 2-frugally (resp. linearly) L-list colourable for any list assignment such that |L(v)| â„ k for all v â 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
Improved bounds on coloring of graphs
Given a graph with maximum degree , we prove that the
acyclic edge chromatic number of is such that . Moreover we prove that:
if has girth ; a'(G)\le
\lceil5.77 (\Delta-1)\rc if
has girth ; a'(G)\le \lc4.52(\D-1)\rc if ;
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 of is
such that
a(G)\le \lc 6.59 \Delta^{4/3}+3.3\D\rc. We also prove that the
star-chromatic number of 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 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
Distributed -Coloring in Sublogarithmic Rounds
We give a new randomized distributed algorithm for -coloring in
the LOCAL model, running in
rounds in a graph of maximum degree~. This implies that the
-coloring problem is easier than the maximal independent set
problem and the maximal matching problem, due to their lower bounds of by Kuhn, Moscibroda, and Wattenhofer [PODC'04].
Our algorithm also extends to list-coloring where the palette of each node
contains colors. We extend the set of distributed symmetry-breaking
techniques by performing a decomposition of graphs into dense and sparse parts
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