13,517 research outputs found
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
On-line list colouring of random graphs
In this paper, the on-line list colouring of binomial random graphs G(n,p) is
studied. We show that the on-line choice number of G(n,p) is asymptotically
almost surely asymptotic to the chromatic number of G(n,p), provided that the
average degree d=p(n-1) tends to infinity faster than (log log n)^1/3(log
n)^2n^(2/3). For sparser graphs, we are slightly less successful; we show that
if d>(log n)^(2+epsilon) for some epsilon>0, then the on-line choice number is
larger than the chromatic number by at most a multiplicative factor of C, where
C in [2,4], depending on the range of d. Also, for d=O(1), the on-line choice
number is by at most a multiplicative constant factor larger than the chromatic
number
On the chromatic number of random geometric graphs
Given independent random points X_1,...,X_n\in\eR^d with common probability
distribution , and a positive distance , we construct a random
geometric graph with vertex set where distinct and
are adjacent when \norm{X_i-X_j}\leq r. Here \norm{.} may be any norm on
\eR^d, and may be any probability distribution on \eR^d with a
bounded density function. We consider the chromatic number of
and its relation to the clique number as . Both
McDiarmid and Penrose considered the range of when and the range when , and their
results showed a dramatic difference between these two cases. Here we sharpen
and extend the earlier results, and in particular we consider the `phase
change' range when with a fixed
constant. Both McDiarmid and Penrose asked for the behaviour of the chromatic
number in this range. We determine constants such that
almost surely. Further, we find a "sharp
threshold" (except for less interesting choices of the norm when the unit ball
tiles -space): there is a constant such that if then
tends to 1 almost surely, but if then
tends to a limit almost surely.Comment: 56 pages, to appear in Combinatorica. Some typos correcte
A tree-decomposed transfer matrix for computing exact Potts model partition functions for arbitrary graphs, with applications to planar graph colourings
Combining tree decomposition and transfer matrix techniques provides a very
general algorithm for computing exact partition functions of statistical models
defined on arbitrary graphs. The algorithm is particularly efficient in the
case of planar graphs. We illustrate it by computing the Potts model partition
functions and chromatic polynomials (the number of proper vertex colourings
using Q colours) for large samples of random planar graphs with up to N=100
vertices. In the latter case, our algorithm yields a sub-exponential average
running time of ~ exp(1.516 sqrt(N)), a substantial improvement over the
exponential running time ~ exp(0.245 N) provided by the hitherto best known
algorithm. We study the statistics of chromatic roots of random planar graphs
in some detail, comparing the findings with results for finite pieces of a
regular lattice.Comment: 5 pages, 3 figures. Version 2 has been substantially expanded.
Version 3 shows that the worst-case running time is sub-exponential in the
number of vertice
- …