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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2509.09407 |
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| _version_ | 1866909781918220288 |
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| author | Huang, Danjun Guo, Yuqian |
| author_facet | Huang, Danjun Guo, Yuqian |
| contents | 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 $χ_i'(G)$. A graph is called claw-free if it has no induced subgraph isomorphic to the complete bipartite graph $K_{1,3}$. In this paper, we show that $χ_i'(G)\le 13$ for every claw-free graph $G$ with $Δ(G)\leq 4$, where $Δ(G)$ is the maximum degree of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09407 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Injective edge-coloring of claw-free graphs with maximum degree 4 Huang, Danjun Guo, Yuqian Combinatorics 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 $χ_i'(G)$. A graph is called claw-free if it has no induced subgraph isomorphic to the complete bipartite graph $K_{1,3}$. In this paper, we show that $χ_i'(G)\le 13$ for every claw-free graph $G$ with $Δ(G)\leq 4$, where $Δ(G)$ is the maximum degree of $G$. |
| title | Injective edge-coloring of claw-free graphs with maximum degree 4 |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.09407 |