关于循环覆盖的色数与色指数的谱
On the Spectra of Chromatic Number and Chromatic Index of Cyclic Covers
AI总结：
该研究分析无环多重图的ℓ重循环覆盖的色数与色指数谱，明确了不同奇偶性ℓ对应的谱范围及判定条件，为图的着色性质提供了新结论。
AI中文摘要：
对于固定整数ℓ≥2，本研究探讨无环多重图的ℓ重循环覆盖可达到的色指数与色数值。针对边着色，首先研究作为色指数基本下界的密度，证明图G的每个ℓ重循环覆盖的密度均不超过G的密度；进一步证明当ℓ为偶数时，G的所有ℓ重循环覆盖的色指数谱包含Δ(G)与χ'(G)之间的所有整数；当ℓ为奇数时，一般情况下色指数谱未必完整，对边色临界图可精确确定其可达值。针对顶点着色，证明若χ(G)≥3，则G的所有ℓ重循环覆盖的色数谱包含3与χ(G)之间的所有整数；此外，该谱包含2当且仅当G为二分图或ℓ为偶数。
英文摘要：
For a fixed integer $\ell \ge 2$, we study what values of chromatic index and chromatic number can be attained by some $\ell$-fold cyclic cover of a loopless multigraph. For edge-coloring, we first investigate the density, a fundamental lower bound for the chromatic index, and show that the density of every $\ell$-fold cyclic cover of a graph $G$ is at most that of $G$. We further prove that if $\ell$ is even, then the spectrum of chromatic indices over all $\ell$-fold cyclic covers of $G$ contains every integer between $Δ(G)$ and $χ'(G)$. When $\ell$ is odd, the chromatic-index spectrum need not be complete in general; for edge-chromatic critical graphs, we determine exactly which values are attainable. For vertex-coloring, we prove that if $χ(G)\ge 3$, then the spectrum of chromatic numbers over all $\ell$-fold cyclic covers of $G$ contains every integer between $3$ and $χ(G)$. Moreover, this spectrum contains $2$ if and only if $G$ is bipartite or $\ell$ is even.
