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Injective edge coloring of sparse graphs with maximum degree 5


A k-injective-edge coloring of a graph G is a mapping \(c:E(G)\rightarrow \{1,2,\cdots ,k\}\) such that \(c(e_1)\ne c(e_3)\) for any three consecutive edges \(e_1,e_2,e_3\) of a path or a 3-cycle. \(\chi _{i}'(G)=\min \{k|G\) has a k-injective-edge coloring\(\}\) is called the injective chromatic index of G. In this paper, we prove that for graphs G with \(\Delta (G)\le 5\), (1) \(\chi _{i}'(G)\le 8\) if \(mad(G)<\frac{7}{3}\); (2) \(\chi _{i}'(G)\le 9\) if \(mad(G)<\frac{12}{5}\); (3) \(\chi _{i}'(G)\le 10\) if \(mad(G)<\frac{5}{2}\); (4) \(\chi _{i}'(G)\le 11\) if \(mad(G)<\frac{18}{7}\).

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Correspondence to Junlei Zhu.

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This research was supported by National Science Foundation of China under Grant Nos. 11901243, 12201569.

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Zhu, J., Bu, Y. & Zhu, H. Injective edge coloring of sparse graphs with maximum degree 5. J Comb Optim 45, 46 (2023).

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  • Maximum degree
  • Maximum average degree
  • Injective edge coloring

Mathematics Subject Classification

  • 05C15