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arXiv · 1912.02467

On star edge colorings of bipartite and subcubic graphs

Carl Johan Casselgren·Jonas B. Granholm·André Raspaud
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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 $χ'_{st}(G)$ 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 ($k\geq 1$), we prove that if $b=2k+1$ then $χ'_{st}(G) \leq 3k+2$; and if $b=2k$, then $χ'_{st}(G) \leq 3k$; 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.

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Carl Johan Casselgren, Jonas B. Granholm, André Raspaud. 2020-01-27. On star edge colorings of bipartite and subcubic graphs. https://doi.org/10.1016/j.dam.2021.03.007

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