哈密顿有向图的两块循环与色数
Two-block cycles and chromatic number of Hamiltonian digraphs
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中文总结 AI 辅助
研究不含\(C(k,\ell)\)的哈密顿有向图的色数问题，通过证明当\(k + \ell \geq 6\)时色数\(\chi(D) \leq k + \ell - 1\)且此界最优，解决了Addario - Berry等人提出的相关问题。
中文摘要 AI 辅助
设\(k\)和\(\ell\)为正整数。族\(C(k,\ell)\)由分别从两条长度至少为\(k\)和\(\ell\)的内部顶点不相交的有向路径，通过识别它们的初始顶点和终端顶点得到的所有有向图组成。Addario - Berry、Havet和Thomassé（2007年《组合论杂志B辑》）提出，对于任意满足\(k + \ell \geq 4\)的正整数\(k\)和\(\ell\)，每个不含\(C(k,\ell)\)的强连通有向图\(D\)的色数\(\chi(D)\)是否至多为\(k + \ell - 1\)。设\(D\)是不含\(C(k,\ell)\)的哈密顿有向图。Kim、Kim、Ma和Park（2018年《图论杂志》）表明\(\chi(D) \leq k + \ell\)，且当\(k + \ell = 5\)时达到该界。本文证明对于\(k + \ell \geq 6\)，\(\chi(D) \leq k + \ell - 1\)，且该界对于所有\(k + \ell \geq 6\)是最优的，解决了Addario - Berry、Havet和Thomassé针对哈密顿有向图提出的问题。
英文摘要
Let $k$ and $\ell$ be positive integers. The family $C(k,\ell)$ consists of all digraphs obtained from two internally vertex-disjoint directed paths of lengths at least $k$ and $\ell$, respectively, and identifying their initial vertices and their terminal vertices. Addario-Berry, Havet and Thomassé (JCT-B, 2007) asked whether, for any positive integers $k$ and $\ell$ with $k+\ell \ge 4$, the chromatic number $χ(D)$ is at most $k+\ell-1$ for every $C(k,\ell)$-free strongly connected digraph $D$. Let $D$ be a $C(k,\ell)$-free Hamiltonian digraph. Kim, Kim, Ma and Park (JGT, 2018) showed that $χ(D) \le k+\ell$ and the bound is attained when $k+\ell=5$. In this paper, we prove that $χ(D) \le k+\ell-1$ for $k+\ell\ge 6$ and this bound is best possible for all $k+\ell\geq 6$, which resolves the problem posed by Addario-Berry, Havet and Thomassé for Hamiltonian digraphs.
