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arXiv · 2509.16733

List Coloring the Cartesian Product of a Complete Graph and Complete Bipartite Graph

Hemanshu Kaul·Leonardo Marciaga·Jeffrey A. Mudrock
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Abstract

We study the list chromatic number of the Cartesian product of a complete graph of order $n$ and a complete bipartite graph with partite sets of size $a$ and $b$, denoted $χ_{\ell}(K_n \square K_{a,b})$. At the 2024 Sparse Graphs Coalition's Workshop on algebraic, extremal, and structural methods and problems in graph colouring, Mudrock presented the following question: For each positive integer $a$, does $χ_{\ell}(K_n \square K_{a,b}) = n+a$ if and only if $b \geq (n+a-1)!^a/(a-1)!^a$? In this paper, we show the answer to this question is yes by studying $χ_{\ell}(H \square K_{a,b})$ when $H$ is strongly chromatic-choosable (a special form of vertex criticality) with the help of the list color function and analytic inequalities such as that of Karamata. Our result can be viewed as a generalization of the well-known result that $χ_{\ell}(K_{a,b}) = 1+a$ if and only if $b \geq a^a$.

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Hemanshu Kaul, Leonardo Marciaga, Jeffrey A. Mudrock. 2025-11-06. List Coloring the Cartesian Product of a Complete Graph and Complete Bipartite Graph. https://arxiv.org/abs/2509.16733

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