7,354 research outputs found
Distance edge-colourings and matchings
AbstractWe consider a distance generalisation of the strong chromatic index and the maximum induced matching number. We study graphs of bounded maximum degree and Erdős–Rényi random graphs. We work in three settings. The first is that of a distance generalisation of an Erdős–Nešetřil problem. The second is that of an upper bound on the size of a largest distance matching in a random graph. The third is that of an upper bound on the distance chromatic index for sparse random graphs. One of our results gives a counterexample to a conjecture of Skupień
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
Strong chromatic index of sparse graphs
A coloring of the edges of a graph is strong if each color class is an
induced matching of . The strong chromatic index of , denoted by
, is the least number of colors in a strong edge coloring
of . In this note we prove that for every -degenerate graph . This confirms the strong
version of conjecture stated recently by Chang and Narayanan [3]. Our approach
allows also to improve the upper bound from [3] for chordless graphs. We get
that for any chordless graph . Both
bounds remain valid for the list version of the strong edge coloring of these
graphs
Local Graph Coloring and Index Coding
We present a novel upper bound for the optimal index coding rate. Our bound
uses a graph theoretic quantity called the local chromatic number. We show how
a good local coloring can be used to create a good index code. The local
coloring is used as an alignment guide to assign index coding vectors from a
general position MDS code. We further show that a natural LP relaxation yields
an even stronger index code. Our bounds provably outperform the state of the
art on index coding but at most by a constant factor.Comment: 14 Pages, 3 Figures; A conference version submitted to ISIT 2013;
typos correcte
- …