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 complete
bipartite graphs; in particular we obtain tight upper bounds for the case when
one part has size at most 3. We also consider 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.
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.Comment: 18 page