<abstract><p>The star chromatic index of a graph $ G $, denoted by $ \chi{'}_{st}(G) $, is the smallest number of colors required to properly color $ E(G) $ such that every connected bicolored subgraph is a path with no more than three edges. A graph is $ K_{2, t} $-free if it contains no $ K_{2, t} $ as a subgraph. This paper proves that every $ K_{2, t} $-free planar graph $ G $ satisfies $ \chi_{st}'(G)\le 1.5\Delta +20t+20 $, which is sharp up to the constant term. In particular, our result provides a common generalization of previous results on star edge coloring of outerplanar graphs by Bezegová et al.(2016) and of $ C_4 $-free planar graphs by Wang et al.(2018), as those graphs are subclasses of $ K_{2, 3} $-free planar graphs.</p></abstract>
A strong edge coloring of a graph G is a proper coloring of edges in G such that any two edges of distance at most 2 are colored with distinct colors. The strong chromatic index χs′(G) is the smallest integer l such that G admits a strong edge coloring using l colors. A K4(t)-minor free graph is a graph that does not contain K4(t) as a contraction subgraph, where K4(t) is obtained from a K4 by subdividing edges exactly t−4 times. The paper shows that every K4(t)-minor free graph with maximum degree Δ(G) has χs′(G)≤(t−1)Δ(G) for t∈{5,6,7} which generalizes some known results on K4-minor free graphs by Batenburg, Joannis de Verclos, Kang, Pirot in 2022 and Wang, Wang, and Wang in 2018. These upper bounds are sharp.
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