49 research outputs found
Star 5-edge-colorings of subcubic multigraphs
The star chromatic index of a multigraph , denoted , is the
minimum number of colors needed to properly color the edges of such that no
path or cycle of length four is bi-colored. A multigraph is star
-edge-colorable if . Dvo\v{r}\'ak, Mohar and \v{S}\'amal
[Star chromatic index, J Graph Theory 72 (2013), 313--326] proved that every
subcubic multigraph is star -edge-colorable, and conjectured that every
subcubic multigraph should be star -edge-colorable. Kerdjoudj, Kostochka and
Raspaud considered the list version of this problem for simple graphs and
proved that every subcubic graph with maximum average degree less than is
star list--edge-colorable. It is known that a graph with maximum average
degree is not necessarily star -edge-colorable. In this paper, we
prove that every subcubic multigraph with maximum average degree less than
is star -edge-colorable.Comment: to appear in Discrete Mathematics. arXiv admin note: text overlap
with arXiv:1701.0410
Extension from precoloured sets of edges
We consider precolouring extension problems for proper edge-colourings of graphs and multigraphs, in an attempt to prove stronger versions of Vizing's and Shannon's bounds on the chromatic index of (multi)graphs in terms of their maximum degree Δ. We are especially interested in the following question: when is it possible to extend a precoloured matching to a colouring of all edges of a (multi)graph? This question turns out to be related to the notorious List Colouring Conjecture and other classic notions of choosability