Abstract:We show that if $G$ is a $d$-regular Vizing-class-1 graph, then the proper additive chromatic index of $G$, denoted $\eta'_p(G)$, is equal to its chromatic index. This verifies that a strengthening of the Additive Coloring Conjecture of Czerwiński et al. holds for line graphs of $d$-regular Vizing-class-1 graphs. We show that if $G$ is a $d$-regular Vizing-class-2 graph, $\eta'_{p}(G)\leq \frac{(2^{\lceil \log_2 (d+1)\rceil})^2+2}{3}$, and if $G$ is a $d$-regular Vizing-class-2 graph that admits a proper edge-coloring with a smallest color class of size $r$ and $\text{girth}(G)\geq 6r-5$, then $\eta_p'(G)\leq 2d$, among other results.
From: Ian Gossett [view email]
[v1]
Tue, 26 May 2026 19:42:40 UTC (249 KB)
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