关于图的模边着色的两个猜想的反例
Counterexamples to two conjectures on modular edge colorings of graphs
浏览论文内容
中文总结 AI 辅助
本文针对图的模边着色领域的两个猜想构造了反例，证明相关猜想不成立，得到了特定 $0_k$-图的模边色数下界，否定了两个关于 $\chi_k'(G)$ 上界的结论。
中文摘要 AI 辅助
对于整数 $k\geq2$，设 $\chi_k'(G)$ 表示图 $G$ 的边着色中满足每个颜色子图的非零度数均模 $k$ 余 1 的最小颜色数。若图的每个顶点度数都能被 $k$ 整除，则称其为 $0_k$-图。我们否定了Berthe等人的猜想（《On modular edge colorings of graphs》，SIAM J. Discrete Math. 40 (2026) 897--904），该猜想称对每个 $0_k$-图 $G$ 有 $\chi_k'(G)\leq k+o(k)$。我们针对度数集为 $\{k,2k\}$ 且具有特定顶点划分的 $0_k$-图证明了一个下界：当各部分大小取合适值时，若某一部分内部的边数为 $o(k^2)$，则 $\chi_k'(G)\geq(4-2\sqrt{2}+o(1))k$。这一结果给出了连通二分图和连通非二分图的反例，且这些反例同时否定了Botler、Colucci和Kohayakawa的早期猜想（《The mod $k$ chromatic index of graphs is $O(k)$》，J. Graph Theory 102 (2023) 197--200），该猜想称存在绝对常数 $C$ 使得 $\chi_k'(G)\leq k+C$。
英文摘要
For an integer $k\geq2$, let $χ_k'(G)$ denote the minimum number of colors in an edge-coloring of a graph $G$ such that every nonzero degree in each color subgraph is congruent to $1\pmod{k}$. A graph is a $0_k$-graph if every vertex degree is divisible by $k$. We disprove a conjecture of Berthe et al.\ (On modular edge colorings of graphs, SIAM J. Discrete Math. 40 (2026) 897--904), which states that $χ_k'(G)\leq k+o(k)$ for every $0_k$-graph $G$. We prove a lower bound for $0_k$-graphs with degree set $\{k,2k\}$ and a specified vertex partition. With a suitable choice of the part sizes, if the number of edges inside one part is $o(k^2)$, then $χ_k'(G)\geq(4-2\sqrt2+o(1))k$. This gives connected bipartite and connected nonbipartite counterexamples. In particular, the same examples also disprove the earlier conjecture of Botler, Colucci, and Kohayakawa (The mod $k$ chromatic index of graphs is $O(k)$, J. Graph Theory 102 (2023) 197--200), which states that $χ_k'(G)\leq k+C$ for some absolute constant $C$.
