1 research outputs found
On the Strong Chromatic Index of Sparse Graphs
The strong chromatic index of a graph , denoted , is the least
number of colors needed to edge-color so that edges at distance at most two
receive distinct colors. The strong list chromatic index, denoted
, is the least integer such that if arbitrary lists of
size are assigned to each edge then can be edge-colored from those
lists where edges at distance at most two receive distinct colors. We use the
discharging method, the Combinatorial Nullstellensatz, and computation to show
that if is a subcubic planar graph with
then , answering a question of Borodin and Ivanova
[Precise upper bound for the strong edge chromatic number of sparse planar
graphs, Discuss. Math. Graph Theory, 33(4), (2014) 759--770]. We further show
that if is a subcubic planar graph and ,
then , improving a bound from the same paper. Finally, if
is a planar graph with maximum degree at most four and
, then , improving a more
general bound of Wang and Zhao from [Odd graphs and its application on the
strong edge coloring, arXiv:1412.8358] in this case.Comment: 15 pages, 10 figure