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Injective Edge Coloring for Graphs with Small Edge Weight

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Abstract

An injective k-edge coloring of a graph \(G=(V(G),E(G))\) is a k-edge coloring \(\varphi \) such that if \(e_1\) and \(e_2\) are at distance exactly 2 or in the same triangle, then \(\varphi (e_1)\ne \varphi (e_2)\). The injective chromatic index of G, denoted by \(\chi _i'(G)\), is the minimum k such that G has an injective k-edge coloring. The edge weight of G is defined as \(ew(G)=\max \{d_G(u)+d_G(v):uv\in E(G)\}\). In this paper, we show that \(\chi _i'(G)\le 3\) if \(ew(G)\le 5\); \(\chi _i'(G)\le 7\) if \(ew(G)\le 6\); and \(\chi _i'(G)\le 11\) if \(ew(G)\le 7\).

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Funding

Huiqing Liu was partially supported by NSFC under Grant Number 11971158. Xiaolan Hu was partially supported by NSFC under Grant Number 11971196 and Hubei Provincial Science and Technology Innovation Base (Platform) Special Project 2020DFH002.

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Authors and Affiliations

  1. Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, 430062, People’s Republic of China

    Jian Lu & Huiqing Liu

  2. School of Mathematics and Statistics, Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, P.O. Box 71010, Wuhan, 430079, People’s Republic of China

    Xiaolan Hu

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  1. Jian Lu
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  2. Huiqing Liu
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  3. Xiaolan Hu
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Correspondence to Xiaolan Hu.

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Lu, J., Liu, H. & Hu, X. Injective Edge Coloring for Graphs with Small Edge Weight. Graphs and Combinatorics 38, 160 (2022). https://doi.org/10.1007/s00373-022-02562-3

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