List star edge coloring of generalized Halin graphs

Abstract

A star k-edge coloring is a proper edge coloring such that there are no bichromatic paths or cycles of length four. The smallest integer k such that G admits a star k-edge coloring is the star chromatic index of G. Deng et al. [5], and Bezegová et al. [1] independently proved that the star chromatic index of a tree is at most ⌊3Δ2⌋, and the bound is sharp. Han et al. [8] strengthened the result to list version of star chromatic index, and proved that ⌊3Δ2⌋ is also the sharp upper bound for the list star chromatic index of trees. A generalized Halin graph is a plane graph that consists of a plane embedding of a tree T with Δ(T)≥3, and a cycle C connecting all the leaves of the tree such that C is the boundary of the exterior face. In this paper, we prove that if H≔T∪C is a generalized Halin graph with |C|≠5, then its list star chromatic index is at mostmax⁡{⌊θ(T)+Δ(T)2⌋,2⌊Δ(T)2⌋+7}, where θ(T)=maxxy∈E(T)⁡{dT(x)+dT(y)}. As a consequence, if H is a (generalized) Halin graph with maximum degree Δ≥13, then the list star chromatic index is at most ⌊3Δ2⌋. Moreover, the upper bound for the list star chromatic index is sharp.