13 research outputs found
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
Distance colouring without one cycle length
We consider distance colourings in graphs of maximum degree at most and
how excluding one fixed cycle length affects the number of colours
required as . For vertex-colouring and , if any two
distinct vertices connected by a path of at most edges are required to be
coloured differently, then a reduction by a logarithmic (in ) factor against
the trivial bound can be obtained by excluding an odd cycle length
if is odd or by excluding an even cycle length . For edge-colouring and , if any two distinct edges connected by
a path of fewer than edges are required to be coloured differently, then
excluding an even cycle length is sufficient for a logarithmic
factor reduction. For , neither of the above statements are possible
for other parity combinations of and . These results can be
considered extensions of results due to Johansson (1996) and Mahdian (2000),
and are related to open problems of Alon and Mohar (2002) and Kaiser and Kang
(2014).Comment: 14 pages, 1 figur
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