Graphs and Combinatorics

, Volume 35, Issue 6, pp 1571–1583

List Coloring a Cartesian Product with a Complete Bipartite Factor

• Hemanshu Kaul
• Jeffrey A. Mudrock
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

Notes

Acknowledgements

The authors would like to thank the two anonymous referees whose comments helped improve the readability of this note.

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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