13,565 research outputs found
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
The set chromatic number of random graphs
In this paper we study the set chromatic number of a random graph
for a wide range of . We show that the set chromatic number, as a
function of , forms an intriguing zigzag shape
The Chromatic Number of Random Regular Graphs
Given any integer d >= 3, let k be the smallest integer such that d < 2k log
k. We prove that with high probability the chromatic number of a random
d-regular graph is k, k+1, or k+2, and that if (2k-1) \log k < d < 2k \log k
then the chromatic number is either k+1 or k+2
The t-improper chromatic number of random graphs
We consider the -improper chromatic number of the Erd{\H o}s-R{\'e}nyi
random graph . The t-improper chromatic number of is
the smallest number of colours needed in a colouring of the vertices in which
each colour class induces a subgraph of maximum degree at most . If ,
then this is the usual notion of proper colouring. When the edge probability
is constant, we provide a detailed description of the asymptotic behaviour
of over the range of choices for the growth of .Comment: 12 page
On the strong chromatic number of random graphs
Let G be a graph with n vertices, and let k be an integer dividing n. G is
said to be strongly k-colorable if for every partition of V(G) into disjoint
sets V_1 \cup ... \cup V_r, all of size exactly k, there exists a proper vertex
k-coloring of G with each color appearing exactly once in each V_i. In the case
when k does not divide n, G is defined to be strongly k-colorable if the graph
obtained by adding k \lceil n/k \rceil - n isolated vertices is strongly
k-colorable. The strong chromatic number of G is the minimum k for which G is
strongly k-colorable. In this paper, we study the behavior of this parameter
for the random graph G(n, p). In the dense case when p >> n^{-1/3}, we prove
that the strong chromatic number is a.s. concentrated on one value \Delta+1,
where \Delta is the maximum degree of the graph. We also obtain several weaker
results for sparse random graphs.Comment: 16 page
On the chromatic number of random Cayley graphs
Let G be an abelian group of cardinality N, where (N,6) = 1, and let A be a
random subset of G. Form a graph Gamma_A on vertex set G by joining x to y if
and only if x + y is in A. Then, almost surely as N tends to infinity, the
chromatic number chi(Gamma_A) is at most (1 + o(1))N/2 log_2 N. This is
asymptotically sharp when G = Z/NZ, N prime.
Presented at the conference in honour of Bela Bollobas on his 70th birthday,
Cambridge August 2013.Comment: 26 pages, revised following referee's repor
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