962 research outputs found
An upper bound on the fractional chromatic number of triangle-free subcubic graphs
An -coloring of a graph is a function which maps the vertices
of into -element subsets of some set of size in such a way that
is disjoint from for every two adjacent vertices and in
. The fractional chromatic number is the infimum of over
all pairs of positive integers such that has an -coloring.
Heckman and Thomas conjectured that the fractional chromatic number of every
triangle-free graph of maximum degree at most three is at most 2.8. Hatami
and Zhu proved that . Lu and Peng improved
the bound to . Recently, Ferguson, Kaiser
and Kr\'{a}l' proved that . In this paper,
we prove that
The game chromatic number of random graphs
Given a graph G and an integer k, two players take turns coloring the
vertices of G one by one using k colors so that neighboring vertices get
different colors. The first player wins iff at the end of the game all the
vertices of G are colored. The game chromatic number \chi_g(G) is the minimum k
for which the first player has a winning strategy. In this paper we analyze the
asymptotic behavior of this parameter for a random graph G_{n,p}. We show that
with high probability the game chromatic number of G_{n,p} is at least twice
its chromatic number but, up to a multiplicative constant, has the same order
of magnitude. We also study the game chromatic number of random bipartite
graphs
Some results on (a:b)-choosability
A solution to a problem of Erd\H{o}s, Rubin and Taylor is obtained by showing
that if a graph is -choosable, and , then is not
necessarily -choosable. Applying probabilistic methods, an upper bound
for the choice number of a graph is given. We also prove that a
directed graph with maximum outdegree and no odd directed cycle is
-choosable for every . Other results presented in this
article are related to the strong choice number of graphs (a generalization of
the strong chromatic number). We conclude with complexity analysis of some
decision problems related to graph choosability
On the expansion constant and distance constrained colourings of hypergraphs
For any two non-negative integers h and k, h > k, an L(h, k)-colouring of a
graph G is a colouring of vertices such that adjacent vertices admit colours
that at least differ by h and vertices that are two distances apart admit
colours that at least differ by k. The smallest positive integer {\delta} such
that G permits an L(h, k)-colouring with maximum colour {\delta} is known as
the L(h, k)-chromatic number (L(h, k)-colouring number) denoted by
{\lambda}_{h,k}(G). In this paper, we discuss some interesting invariants in
hypergraphs. In fact, we study the relation between the spectral gap and L(2,
1)-chromatic number of hypergraphs. We derive some inequalities which relates
L(2, 1)-chromatic number of a k-regular simple graph to its spectral gap and
expansion constant. The upper bound of L(h, k)-chromatic number in terms of
various hypergraph invariants such as strong chromatic number, strong
independent number and maximum degree is obtained. We determine the sharp upper
bound for L(2, 1)-chromatic number of hypertrees in terms of its maximum
degree. Finally, we conclude this paper with a discussion on L(2, 1)-colouring
in cartesian product of some classes of hypergraphs
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