18 research outputs found
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
Proper Conflict-free Coloring of Graphs with Large Maximum Degree
A proper coloring of a graph is conflict-free if, for every non-isolated
vertex, some color is used exactly once on its neighborhood. Caro,
Petru\v{s}evski, and \v{S}krekovski proved that every graph has a proper
conflict-free coloring with at most colors and conjectured that
colors suffice for every connected graph with . Our first main result is that even for list-coloring, colors suffice for every graph
with ; we also prove slightly weaker bounds for all
graphs with . These results follow from our more general
framework on proper conflict-free list-coloring of a pair consisting of a graph
and a ``conflict'' hypergraph . As another corollary of our
results in this general framework, every graph has a proper
-list-coloring such that every bi-chromatic
component is a path on at most three vertices, where the number of colors is
optimal up to a constant factor. Our proof uses a fairly new type of recursive
counting argument called Rosenfeld counting, which is a variant of the
Lov\'{a}sz Local Lemma or entropy compression.
We also prove an asymptotically optimal result for a fractional analogue of
our general framework for proper conflict-free coloring for pairs of a graph
and a conflict hypergraph. A corollary states that every graph has a
fractional -coloring such that every fractionally
bi-chromatic component has at most two vertices. In particular, it implies that
the fractional analogue of the conjecture of Caro et al. holds asymptotically
in a strong sense
New bounds for odd colourings of graphs
Given a graph , a vertex-colouring of , and a subset
, a colour is said to be \emph{odd} for
in if it has an odd number of occurrences in . We say that
is an \emph{odd colouring} of if it is proper and every (open)
neighbourhood has an odd colour in . The odd chromatic number of a
graph , denoted by , is the minimum such that an
odd colouring exists. In a recent paper, Caro,
Petru\v sevski and \v Skrekovski conjectured that every connected graph of
maximum degree has odd-chromatic number at most . We
prove that this conjecture holds asymptotically: for every connected graph
with maximum degree , as . We also prove that for every
. If moreover the minimum degree of is sufficiently large,
we have and . Finally, given an integer , we study the
generalisation of these results to -odd colourings, where every vertex
must have at least odd colours in its neighbourhood. Many
of our results are tight up to some multiplicative constant
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
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