Injective edge-coloring of claw-free graphs with maximum degree 4

Danjun Huang (Zhejiang Normal University)
Yuqian Guo (Zhejiang Normal University)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 14, No 2 (2026): Electronic Journal of Graph Theory and Applications

An injective k-edge-coloring of a graph G is a mapping ϕ : E(G) → {1, 2, …, k}, such that ϕ(e) ≠ ϕ(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 K1,3. In this paper, we show that χ′i(G) ≤ 13 for every claw-free graph G with Δ(G) ≤ 4, where Δ(G) is the maximum degree of G.