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Sharp upper bound of injective coloring of planar graphs with girth at least 5

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Abstract

An injective k-coloring of a graph G is a k-coloring c (not necessarily proper) such that \(c(u)\ne c(v)\) whenever u, v has a common neighbor in G. The injective chromatic number of G, denoted by \(\chi _i(G)\), is the least integer k such that G has an injective k-coloring. We prove that the injective chromatic number of planar graphs with \(g \ge 5\) and \(\Delta \ge 2339\) is at most \(\Delta + 1\), and this bound is sharp.

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  1. School of Mathematical Sciences, Tongji University, Shanghai, 200092, China

    Qiming Fang & Li Zhang

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  1. Qiming Fang
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  2. Li Zhang
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All authors contributed to the study conception and design. The first draft of the manuscript was written by Qiming Fang and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Qiming Fang or Li Zhang.

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Supported by the Natural Science Foundation of China, Grant No.11871377.

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Fang, Q., Zhang, L. Sharp upper bound of injective coloring of planar graphs with girth at least 5. J Comb Optim 44, 1161–1198 (2022). https://doi.org/10.1007/s10878-022-00880-z

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