6 research outputs found
Strong chromatic index of k-degenerate graphs
A {\em strong edge coloring} of a graph is a proper edge coloring in
which every color class is an induced matching. The {\em strong chromatic
index} \chiup_{s}'(G) of a graph is the minimum number of colors in a
strong edge coloring of . In this note, we improve a result by D{\k e}bski
\etal [Strong chromatic index of sparse graphs, arXiv:1301.1992v1] and show
that the strong chromatic index of a -degenerate graph is at most
. As a direct consequence, the strong
chromatic index of a -degenerate graph is at most ,
which improves the upper bound by Chang and Narayanan
[Strong chromatic index of 2-degenerate graphs, J. Graph Theory 73 (2013) (2)
119--126]. For a special subclass of -degenerate graphs, we obtain a better
upper bound, namely if is a graph such that all of its -vertices
induce a forest, then \chiup_{s}'(G) \leq 4 \Delta(G) -3; as a corollary,
every minimally -connected graph has strong chromatic index at most . Moreover, all the results in this note are best possible in
some sense.Comment: 3 pages in Discrete Mathematics, 201
The strong chromatic index of 1-planar graphs
The chromatic index of a graph is the smallest for which
admits an edge -coloring such that any two adjacent edges have distinct
colors. The strong chromatic index of is the smallest such
that has a proper edge -coloring with the condition that any two edges
at distance at most 2 receive distinct colors. A graph is 1-planar if it can be
drawn in the plane so that each edge is crossed by at most one other edge.
In this paper, we show that every graph with maximum average degree
has . As a corollary, we
prove that every 1-planar graph with maximum degree has
, which improves a result, due to Bensmail et
al., which says that if