Spartan Bipartite Graphs are Essentially Elementary

We study a two-player game on a graph between an attacker and a defender. To begin with, the defender places guards on a subset of vertices. In each move, the attacker attacks an edge. The defender must move at least one guard across the attacked edge to defend the attack. The defender wins if and only if the defender can defend an infinite sequence of attacks. The smallest number of guards with which the defender has a winning strategy is called the eternal vertex cover number of a graph $G$ and is denoted by $evc(G)$. It is clear that $evc(G)$ is at least $mvc(G)$, the size of a minimum vertex cover of $G$. We say that $G$ is Spartan if $evc(G) = mvc(G)$. The characterization of Spartan graphs has been largely open. In the setting of bipartite graphs on $2n$ vertices where every edge belongs to a perfect matching, an easy strategy is to have $n$ guards that always move along perfect matchings in response to attacks. We show that these are essentially the only Spartan bipartite graphs.