8,381 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
From light edges to strong edge-colouring of 1-planar graphs
International audienceA strong edge-colouring of an undirected graph is an edge-colouring where every two edges at distance at most~ receive distinct colours. The strong chromatic index of is the least number of colours in a strong edge-colouring of . A conjecture of Erd\H{o}s and Ne\v{s}et\v{r}il, stated back in the 's, asserts that every graph with maximum degree should have strong chromatic index at most roughly . Several works in the last decades have confirmed this conjecture for various graph classes. In particular, lots of attention have been dedicated to planar graphs, for which the strong chromatic index decreases to roughly , and even to smaller values under additional structural requirements.In this work, we initiate the study of the strong chromatic index of -planar graphs, which are those graphs that can be drawn on the plane in such a way that every edge is crossed at most once. We provide constructions of -planar graphs with maximum degree~ and strong chromatic index roughly . As an upper bound, we prove that the strong chromatic index of a -planar graph with maximum degree is at most roughly (thus linear in ). The proof of this result is based on the existence of light edges in -planar graphs with minimum degree at least~
The distance-t chromatic index of graphs
We consider two graph colouring problems in which edges at distance at most
are given distinct colours, for some fixed positive integer . We obtain
two upper bounds for the distance- chromatic index, the least number of
colours necessary for such a colouring. One is a bound of (2-\eps)\Delta^t
for graphs of maximum degree at most , where \eps is some absolute
positive constant independent of . The other is a bound of (as ) for graphs of maximum degree at most
and girth at least . The first bound is an analogue of Molloy and Reed's
bound on the strong chromatic index. The second bound is tight up to a constant
multiplicative factor, as certified by a class of graphs of girth at least ,
for every fixed , of arbitrarily large maximum degree , with
distance- chromatic index at least .Comment: 14 pages, 2 figures; to appear in Combinatorics, Probability and
Computin
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
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