Multithreshold multipartite graphs with small parts

Abstract

A graph is a $k$-threshold graph with thresholds $\theta_1, \theta_2, \dots, \theta_k$ if we can assign a real number $r_v$ to each vertex $v$ such that for any two distinct vertices $u$ and $v$, $uv$ is an edge if and only if the number of thresholds not exceeding $r_u+r_v$ is odd. The threshold number of a graph is the smallest $k$ for which it is a $k$-threshold graph. Multithreshold graphs were introduced by Jamison and Sprague as a generalization of classical threshold graphs. They asked for the exact threshold numbers of complete multipartite graphs. Recently, Chen and Hao solved the problem for complete multipartite graphs where each part is not too small, and they asked for the case when each part has size $3$. We determine the exact threshold numbers of $K_{3, 3, \dots, 3}$, $K_{4, 4, \dots, 4}$ and their complements $nK_3$, $nK_4$. This improves a result of Puleo.