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We study a two-player game on a graph between an attacker and a defender. Tobegin 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 acrossthe attacked edge to defend the attack. The defender wins if and only if thedefender can defend an infinite sequence of attacks. The smallest number ofguards with which the defender has a winning strategy is called the eternalvertex 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 $G$. Wesay that $G$ is Spartan if $evc(G) = mvc(G)$. The characterization of Spartangraphs 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 tohave $n$ guards that always move along perfect matchings in response toattacks. We show that these are essentially the only Spartan bipartite graphs.

Spartan Bipartite Graphs are Essentially Elementary