Abstract
A star edge coloring of a graph is a proper edge coloring with no 2-colored path or cycle of length four. The star chromatic index χst′(G) of G is the minimum number t for which G has a star edge coloring with t colors. We prove upper bounds for the star chromatic index of bipartite graphs G where all vertices in one part have maximum degree 2 and all vertices in the other part has maximum degree b. Let k be an integer (k≥1); we prove that if b=2k+1, then χst′(G)≤3k+2; and if b=2k, then χst′(G)≤3k; both upper bounds are sharp. We also consider complete bipartite graphs; in particular we determine the star chromatic index of such graphs when one part has size at most 3, and prove upper bounds for the general case.Finally, we consider the well-known conjecture that subcubic graphs have star chromatic index at most 6; in particular we settle this conjecture for cubic Halin graphs.
Highlights
A star edge coloring of a graph is a proper edge coloring with no 2-colored path or cycle of length four
The star chromatic index χs′t (G) of G is the minimum number t for which G has a star edge coloring with t colors
We prove upper bounds for the star chromatic index of bipartite graphs G where all vertices in one part have maximum degree 2 and all vertices in the other part has maximum degree b
Summary
A star edge coloring of a graph is a proper edge coloring with no 2-colored path or cycle of length four. Dvořák et al [4] studied star edge colorings of complete graphs and obtained the currently best upper and lower bounds for the star chromatic index of such graphs. Wang et al [19,20] quite recently obtained some upper bounds on the star chromatic index of graphs with maximum degree four, and for some families of planar and related classes of graphs. As for complete graphs, a fundamental problem for complete bipartite graphs is to determine whether the star chromatic index is a linear function on the number of vertices. We obtain analogous results for star edge colorings: we obtain a sharp upper bound for the star chromatic index of a bipartite graph where one part has maximum degree two. In this paper we verify that the conjecture holds for some families of graphs with maximum degree three, namely bipartite graphs where one part has maximum degree 2, cubic Halin graphs and another family of planar graphs
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