A ring is a graph $R$ whose vertex set can be partitioned into $k \geq 4$ nonempty sets, $X_1, \dots, X_k$, such that for all $i \in \{1,\dots,k\}$ the set $X_i$ can be ordered as $X_i = \{u_i^1, \dots, u_i^{|X_i|}\}$ so that $X_i \subseteq N_R[u_i^{|X_i|}] \subseteq \dots \subseteq N_R[u_i^1] = X_{i-1} \cup X_i \cup X_{i+1}$. A hyperhole is a ring $R$ such that for all $i \in \{1,\dots,k\}$, $X_i$ is complete to $X_{i-1}\cup X_{i+1}$. In this paper, we prove that the chromatic number of a ring $R$ is equal to the chromatic number of a maximum hyperhole in $R$. Using this result, we give a polynomial-time coloring algorithm for rings. Rings appeared as one of the basic classes in a decomposition theorem for a class of graphs studied by Boncompagni, Penev, and Vu\v{s}kovi\'c in [Journal of Graph Theory 91 (2019), 192-246]. Using our coloring algorithm for rings, we show that graphs in this larger class can also be colored in polynomial time. Furthermore, we obtain an optimal $\chi$-bounding function for this larger class of graphs, and we also verify Hadwiger's conjecture for it.

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着色环

环是一个图 $R$，其顶点集可以划分为 $k \geq 4$ 个非空集，$X_1, \dots, X_k$，使得对于所有 $i \in \{1,\dots,k\ }$ 集合 $X_i$ 可以排序为 $X_i = \{u_i^1, \dots, u_i^{|X_i|}\}$ 这样 $X_i \subseteq N_R[u_i^{|X_i|}] \ subseteq \dots \subseteq N_R[u_i^1] = X_{i-1} \cup X_i \cup X_{i+1}$。超孔是一个环 $R$ 使得对于所有 $i \in \{1,\dots,k\}$，$X_i$ 到 $X_{i-1}\cup X_{i+1}$ 是完整的. 在本文中，我们证明了环 $R$ 的色数等于 $R$ 中最大超孔的色数。使用这个结果，我们给出了环的多项式时间着色算法。环是 Boncompagni、Penev 和 Vu\v{s}kovi\'c 在 [图论杂志 91 (2019), 192-246] 研究的一类图的分解定理中的基本类之一。使用我们的环着色算法，我们表明这个较大类中的图也可以在多项式时间内着色。此外，我们获得了这一更大类图的最优 $\chi$-bounding 函数，并且我们还验证了 Hadwiger 的猜想。