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arXiv · 2501.04953

Injective edge-coloring of graphs with small maximum degree

Danjun Huang·Yuqian Guo
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Abstract

An injective $k$-edge-coloring of a graph $G$ is a mapping $ϕ$: $E(G)\rightarrow\{1,2,...,k\}$, such that $ϕ(e)\neϕ(e')$ if edges $e$ and $e'$ are at distance two, or are in a triangle. The smallest integer $k$ such that $G$ has an injective $k$-edge-coloring is called the injective chromatic index of $G$, denoted by $χ_i'(G)$. In this paper, we prove that $χ_i'(G)\le 7$ for every graph $G$ with $Δ(G)\leq 4$ and mad$(G)<\frac{8}{3}$, where $Δ(G)$ is the maximum degree of $G$.

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Danjun Huang, Yuqian Guo. 2025-09-11. Injective edge-coloring of graphs with small maximum degree. https://arxiv.org/abs/2501.04953

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