1,487 research outputs found
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
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
New spectral bounds on the chromatic number encompassing all eigenvalues of the adjacency matrix
The purpose of this article is to improve existing lower bounds on the
chromatic number chi. Let mu_1,...,mu_n be the eigenvalues of the adjacency
matrix sorted in non-increasing order.
First, we prove the lower bound chi >= 1 + max_m {sum_{i=1}^m mu_i / -
sum_{i=1}^m mu_{n-i+1}} for m=1,...,n-1. This generalizes the Hoffman lower
bound which only involves the maximum and minimum eigenvalues, i.e., the case
. We provide several examples for which the new bound exceeds the {\sc
Hoffman} lower bound.
Second, we conjecture the lower bound chi >= 1 + S^+ / S^-, where S^+ and S^-
are the sums of the squares of positive and negative eigenvalues, respectively.
To corroborate this conjecture, we prove the weaker bound chi >= S^+/S^-. We
show that the conjectured lower bound is tight for several families of graphs.
We also performed various searches for a counter-example, but none was found.
Our proofs rely on a new technique of converting the adjacency matrix into
the zero matrix by conjugating with unitary matrices and use majorization of
spectra of self-adjoint matrices.
We also show that the above bounds are actually lower bounds on the
normalized orthogonal rank of a graph, which is always less than or equal to
the chromatic number. The normalized orthogonal rank is the minimum dimension
making it possible to assign vectors with entries of modulus one to the
vertices such that two such vectors are orthogonal if the corresponding
vertices are connected.
All these bounds are also valid when we replace the adjacency matrix A by W *
A where W is an arbitrary self-adjoint matrix and * denotes the Schur product,
that is, entrywise product of W and A
Logical limit laws for minor-closed classes of graphs
Let be an addable, minor-closed class of graphs. We prove that
the zero-one law holds in monadic second-order logic (MSO) for the random graph
drawn uniformly at random from all {\em connected} graphs in on
vertices, and the convergence law in MSO holds if we draw uniformly at
random from all graphs in on vertices. We also prove analogues
of these results for the class of graphs embeddable on a fixed surface,
provided we restrict attention to first order logic (FO). Moreover, the
limiting probability that a given FO sentence is satisfied is independent of
the surface . We also prove that the closure of the set of limiting
probabilities is always the finite union of at least two disjoint intervals,
and that it is the same for FO and MSO. For the classes of forests and planar
graphs we are able to determine the closure of the set of limiting
probabilities precisely. For planar graphs it consists of exactly 108
intervals, each of length . Finally, we analyse
examples of non-addable classes where the behaviour is quite different. For
instance, the zero-one law does not hold for the random caterpillar on
vertices, even in FO.Comment: minor changes; accepted for publication by JCT
The biased odd cycle game
In this paper we consider biased Maker-Breaker games played on the edge set
of a given graph . We prove that for every and large enough ,
there exists a constant for which if and
, then Maker can build an odd cycle in the game for
. We also consider the analogous game where Maker and
Breaker claim vertices instead of edges. This is a special case of the
following well known and notoriously difficult problem due to Duffus, {\L}uczak
and R\"{o}dl: is it true that for any positive constants and , there
exists an integer such that for every graph , if , then
Maker can build a graph which is not -colorable, in the
Maker-Breaker game played on the vertices of ?Comment: 10 page
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