Upper Bounds on List Star Chromatic Index of Sparse Graphs

Abstract

A star k-edge-coloring is a proper k-edge-coloring such that every connected bicolored subgraph is a path of length at most 3. The star chromatic index $$\chi _{st}^\prime $$ (G) of a graph G is the smallest integer k such that G has a star k-edge-coloring. The list star chromatic index $$ch_{st}^\prime $$ (G) is defined analogously. The star edge coloring problem is known to be NP-complete, and it is even hard to obtain tight upper bound as it is unknown whether the star chromatic index for complete graph is linear or super linear. In this paper, we study, in contrast, the best linear upper bound for sparse graph classes. We show that for every e > 0 there exists a constant c(e) such that if mad(G) < $${8 \over 3} - \varepsilon $$, then $$ch_{st}^\prime $$ (G) ≤ $${{3{\rm{\Delta }}} \over 2} + c\left( \varepsilon \right)$$ and the coefficient $${3 \over 2}$$ of Δ is the best possible. The proof applies a newly developed coloring extension method by assigning color sets with different sizes.