Proper Additive Edge Colorings of Regular Graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917538069217280 |
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| author | Gossett, Ian |
| author_facet | Gossett, Ian |
| contents | We show that if $G$ is a $d$-regular Vizing-class-1 graph, then the proper additive chromatic index of $G$, denoted $η'_p(G)$, 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, $η'_{p}(G)\leq \frac{(2^{\lceil \log_2 (d+1)\rceil})^2+2}{3}$, 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 $η_p'(G)\leq 2d$, among other results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_27624 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Proper Additive Edge Colorings of Regular Graphs Gossett, Ian Combinatorics We show that if $G$ is a $d$-regular Vizing-class-1 graph, then the proper additive chromatic index of $G$, denoted $η'_p(G)$, 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, $η'_{p}(G)\leq \frac{(2^{\lceil \log_2 (d+1)\rceil})^2+2}{3}$, 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 $η_p'(G)\leq 2d$, among other results. |
| title | Proper Additive Edge Colorings of Regular Graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.27624 |