Graphs and Combinatorics

, Volume 35, Issue 6, pp 1571–1583

# List Coloring a Cartesian Product with a Complete Bipartite Factor

Original Paper

## Abstract

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 $$\chi _\ell (G \square K_{a,b})$$. We have two motivations. A classic result on the gap between list chromatic number and the chromatic number tells us $$\chi _\ell (K_{a,b}) = 1 + a$$ if and only if $$b \ge a^a$$. Since $$\chi _\ell (K_{a,b}) \le 1 + a$$ for any $$b \in {\mathbb {N}}$$, this result tells us the values of b for which $$\chi _\ell (K_{a,b})$$ is as large as possible and far from $$\chi (K_{a,b})=2$$. In this paper we seek to understand when $$\chi _\ell (G \square K_{a,b})$$ is far from $$\chi (G \square K_{a,b}) = \max \{\chi (G), 2 \}$$. It is easy to show $$\chi _\ell (G \square K_{a,b}) \le \chi _\ell (G) + a$$. Borowiecki et al. (Discrete Math 306:1955–1958, 2006) showed that this bound is attainable if b is sufficiently large; specifically, $$\chi _\ell (G \square K_{a,b}) = \chi _\ell (G) + a$$ whenever $$b \ge (\chi _\ell (G) + a - 1)^{a|V(G)|}$$. Given any graph G and $$a \in {\mathbb {N}}$$, we wish to determine the smallest b such that $$\chi _\ell (G \square K_{a,b}) = \chi _\ell (G) + a$$. 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 for some large families of chromatic-choosable graphs.

## Keywords

Graph coloring List coloring Cartesian product List color function Chromatic choosable

05C15

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© Springer Japan KK, part of Springer Nature 2019

## Authors and Affiliations

• Hemanshu Kaul
• 1
• Jeffrey A. Mudrock
• 2
1. 1.Department of Applied MathematicsIllinois Institute of TechnologyChicagoUSA
2. 2.Department of MathematicsCollege of Lake CountyGrayslakeUSA