96 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
On star edge colorings of bipartite and subcubic graphs
A star edge coloring of a graph is a proper edge coloring with no -colored
path or cycle of length four. The star chromatic index of
is the minimum number for which has a star edge coloring with
colors. We prove upper bounds for the star chromatic index of complete
bipartite graphs; in particular we obtain tight upper bounds for the case when
one part has size at most . We also consider bipartite graphs where all
vertices in one part have maximum degree and all vertices in the other part
has maximum degree . Let be an integer (), we prove that if
then ; and if , then ; both upper bounds are sharp.
Finally, we consider the well-known conjecture that subcubic graphs have star
chromatic index at most ; in particular we settle this conjecture for cubic
Halin graphs.Comment: 18 page
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