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arXiv 1811.02420math.CO
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具有完全二分因子的笛卡儿积的列表着色

List Coloring a Cartesian Product with a Complete Bipartite Factor

Hemanshu Kaul, Jeffrey A. Mudrock

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12 pages
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摘要：

本文研究了图$G$与完全二部图的笛卡儿积的列表色数，其中二部图的部集分别为$a$和$b$，记为$χ_\ell(G \square K_{a,b})$ .我们有两个动机。关于表色数与色数的间隔的一个经典结果告诉我们$χ_\ell(K_{a,b}) = 1 + a$当且仅当$b \geq a^a$。由于任何$b \in \mathbb{N}$的$χ_\ell(K_{a,b}) \leq 1 + a$，这个结果告诉我们$b$的值，$χ_\ell(K_{a,b})$尽可能大，远离$χ(K_{a,b})=2$。在本文中，我们试图了解$χ_\ell(G \square K_{a,b})$是远离$χ(G \square K_{a,b}) = \max \{χ(G), 2 \}$。很容易显示$χ_\ell(G \square K_{a,b}) \leq χ_\ell (G) + a$。在2006年，Borowiecki、Jendrol、Král和Miskuf证明，如果$b$足够大，这个界限是可以达到的;具体地说，只要$b \geq (χ_\ell(G) + a - 1)^{a|V(G)|}$，就可以达到$χ_\ell(G \square K_{a,b}) = χ_\ell (G) + a$。给定任何图$G$和$a \in \mathbb{N}$，我们希望确定最小的$b$，使得$χ_\ell(G \square K_{a,b}) = χ_\ell (G) + a$。在本文中，我们表明，列表颜色函数，一个列表模拟的色多项式，提供了正确的概念和工具，在这个问题上取得进展。使用列表颜色函数，我们证明了Borowiecki等人的一般改进。的2006年的结果，我们计算最小的$b$正是一些大家庭的色选择图。

英文摘要：

We study the list chromatic number of the Cartesian product of any graph $G$ and a complete bipartite graph with partite sets of size $a$ and $b$, denoted $χ_\ell(G \square K_{a,b})$. We have two motivations. A classic result on the gap between list chromatic number and the chromatic number tells us $χ_\ell(K_{a,b}) = 1 + a$ if and only if $b \geq a^a$. Since $χ_\ell(K_{a,b}) \leq 1 + a$ for any $b \in \mathbb{N}$, this result tells us the values of $b$ for which $χ_\ell(K_{a,b})$ is as large as possible and far from $χ(K_{a,b})=2$. In this paper we seek to understand when $χ_\ell(G \square K_{a,b})$ is far from $χ(G \square K_{a,b}) = \max \{χ(G), 2 \}$. It is easy to show $χ_\ell(G \square K_{a,b}) \leq χ_\ell (G) + a$. In 2006, Borowiecki, Jendrol, Král, and Miskuf showed that this bound is attainable if $b$ is sufficiently large; specifically, $χ_\ell(G \square K_{a,b}) = χ_\ell (G) + a$ whenever $b \geq (χ_\ell(G) + a - 1)^{a|V(G)|}$. Given any graph $G$ and $a \in \mathbb{N}$, we wish to determine the smallest $b$ such that $χ_\ell(G \square K_{a,b}) = χ_\ell (G) + a$. In this paper we show that the list color function, a list analogue of the chromatic polynomial, provides the right concept and tool for making progress on this problem. Using the list color function, we prove a general improvement on Borowiecki et al.'s 2006 result, and we compute the smallest such $b$ exactly for some large families of chromatic-choosable graphs.

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