On the Alon–Tarsi number of semi-strong product of graphs

Abstract

The Alon–Tarsi number was defined by Jensen and Toft (Graph coloring problems, Wiley, New York, 1995). The Alon–Tarsi number AT(G) of a graph G is the smallest integer k such that G has an orientation D with maximum outdegree \(k-1\) and the number of even circulation is not equal to that of odd circulations in D. It is known that \(\chi (G)\le \chi _l(G)\le AT(G)\) for any graph G, where \(\chi (G)\) and \(\chi _l(G)\) are the chromatic number and the list chromatic number of G. Denote by \(H_1 \square H_2\) and \(H_1\bowtie H_2\) the Cartesian product and the semi-strong product of two graphs \(H_1\) and \(H_2\), respectively. Kaul and Mudrock (Electron J Combin 26(1):P1.3, 2019) proved that \(AT(C_{2k+1}\square P_n)=3\). Li, Shao, Petrov and Gordeev (Eur J Combin 103697, 2023) proved that \(AT(C_n\square C_{2k})=3\) and \(AT(C_{2m+1}\square C_{2n+1})=4\). Petrov and Gordeev (Mosc. J. Comb. Number Theory 10(4):271–279, 2022) proved that \(AT(K_n\square C_{2k})=n\). Note that the semi-strong product is noncommutative. In this paper, we determine \(AT(P_m \bowtie P_n)\), \(AT(C_m \bowtie C_{2n})\), \(AT(C_m \bowtie P_n)\) and \(AT(P_m \bowtie C_{n})\). We also prove that \(5\le AT(C_m \bowtie C_{2n+1})\le 6\).