The Alon-Tarsi Number of Cartesian product and Corona product of Hypercube Graph and Special Graphs

TL;DR

This work addresses the Alon-Tarsi number $AT(G)$ for the hypercube $Q_n$ and its Cartesian and Corona products with various graphs. It uses orientation-construction techniques, subgraph-average arguments, and product-structure lemmas to derive exact values, notably $AT(Q_n)=\lceil\frac{n}{2}\rceil+1$, and explicit formulas for $AT(Q_n\square T_m)$, $AT(Q_n\square C_{2k})$, and $AT(Q_n\circ G_2)$ when $AT(G_2)=2$, along with $AT(Q_n\circ C_m)$ for even/odd cycles. The results show these product graphs are not chromatic-AT-choosable and provide precise values that extend understanding of AT-choosability in graph products. The findings offer new insights into how Cartesian and Corona operations interact with Alon-Tarsi theory on hypercube-related graphs.

Abstract

The \emph{Alon-Tarsi number} of a graph $G$ is the smallest $k$ so that there exists an orientation $D$ of $G$ with max outdegree $k-1$ satisfying the number of even Eulerian subgraphs different from the number of odd Eulerian subgraphs. In this paper, the Alon-Tarsi number of the $n$-cube is obtained according to its special properties, we obtain the Alon-Tarsi number of Cartesian product of some special bipartite graphs, and get the Alon-Tarsi number of Corona product of graphs. As corollaries, we get the Alon-Tarsi number of Cartesian product and Corona product of hypercube graph and special graphs.

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Theorems & Definitions (21)

• Definition 2.1
• Definition 2.2
• Lemma 2.1
• Lemma 2.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof