4,200 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
On the quantum chromatic number of a graph
We investigate the notion of quantum chromatic number of a graph, which is
the minimal number of colours necessary in a protocol in which two separated
provers can convince an interrogator with certainty that they have a colouring
of the graph.
After discussing this notion from first principles, we go on to establish
relations with the clique number and orthogonal representations of the graph.
We also prove several general facts about this graph parameter and find large
separations between the clique number and the quantum chromatic number by
looking at random graphs.
Finally, we show that there can be no separation between classical and
quantum chromatic number if the latter is 2, nor if it is 3 in a restricted
quantum model; on the other hand, we exhibit a graph on 18 vertices and 44
edges with chromatic number 5 and quantum chromatic number 4.Comment: 7 pages, 1 eps figure; revtex4. v2 has some new references; v3 furthe
small improvement
A new upper bound on the game chromatic index of graphs
We study the two-player game where Maker and Breaker alternately color the
edges of a given graph with colors such that adjacent edges never get
the same color. Maker's goal is to play such that at the end of the game, all
edges are colored. Vice-versa, Breaker wins as soon as there is an uncolored
edge where every color is blocked. The game chromatic index
denotes the smallest for which Maker has a winning strategy.
The trivial bounds hold for every
graph , where is the maximum degree of . In 2008, Beveridge,
Bohman, Frieze, and Pikhurko proved that for every there exists a
constant such that holds for any graph
with , and conjectured that the same
holds for every graph . In this paper, we show that is true for all graphs with . In
addition, we consider a biased version of the game where Breaker is allowed to
color edges per turn and give bounds on the number of colors needed for
Maker to win this biased game.Comment: 17 page
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
- …