图的交错多色数
The interleaved multichromatic number of a graph
摘要：
对于$k\ge 1$，我们考虑图的节点的交错$k$元组着色，即，分配$k$不同的自然数给每个节点，以这样的方式，由边连接的节点接收的数字是严格交替的关系$<$。如果至少需要$χ_{int}^k(G)$不同的数来提供具有这样的着色的图$G$，则$G$的交错多色数是$χ_{int}^*(G)=\inf_{k\ge 1}χ_{int}^k(G)/k$，并且已知如果$G$是连通的，则由$G$在非循环方向下的简单圈的函数给出[1]。本文给出了这一结果的一个新的证明。与原来的证明不同，新的证明没有对$G$的连通性做任何假设，也没有求助于交织$k$元组着色及其性质的可能应用。
英文摘要：
For $k\ge 1$, we consider interleaved $k$-tuple colorings of the nodes of a graph, that is, assignments of $k$ distinct natural numbers to each node in such a way that nodes that are connected by an edge receive numbers that are strictly alternating between them with respect to the relation $<$. If it takes at least $χ_{int}^k(G)$ distinct numbers to provide graph $G$ with such a coloring, then the interleaved multichromatic number of $G$ is $χ_{int}^*(G)=\inf_{k\ge 1}χ_{int}^k(G)/k$ and is known to be given by a function of the simple cycles of $G$ under acyclic orientations if $G$ is connected [1]. This paper contains a new proof of this result. Unlike the original proof, the new proof makes no assumptions on the connectedness of $G$, nor does it resort to the possible applications of interleaved $k$-tuple colorings and their properties.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
