List Coloring the Cartesian Product of a Complete Graph and Complete Bipartite Graph
Hemanshu Kaul, Leonardo Marciaga, Jeffrey A. Mudrock
TL;DR
This work resolves the exact threshold for the list-chromatic number of the Cartesian product $K_n \square K_{a,b}$. By leveraging strong chromatic-choosability of $K_n$, the list color function, and a suite of analytic tools (including Karamata's inequality and AM-GM), the authors prove that $\chi_\ell(K_n \square K_{a,b}) = n+a$ if and only if $b \ge \left( \frac{(n+a-1)!}{(a-1)!} \right)^a$. The method combines general upper bounds on $f_a(G)$ with a tight lower-bound construction that hinges on disjoint per-vertex lists and a careful coloring-counting framework, culminating in a precise threshold that generalizes the classical result for $K_{a,b}$. The findings provide a robust approach for analyzing $f_a$ on strongly chromatic-choosable graphs and contribute a significant open-question resolution from the 2024 Sparse Graphs Workshop, with potential applications to broader product-choosability problems.
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 $b \geq a^a$.