A Two-Person Game on Graphs Where Each Player Tries to Encircle his Opponent's Men

We present results on a combinatorial game which was proposed to one of the authors by Ingo Althofer (personal communication). Let G be an undirected finite graph without loops and multiple edges and let k be a positive integer with k ⩽ 12(¦G¦ - 1). There are two players, called white and black, both having k men of their color. In turn, beginning with white, the players position their men one at a time on unoccupied vertices of G. When all men are placed, the players take turns moving a man of their color along an edge to an unoccupied adjacent vertex (again beginning with white). A player wins if his opponent cannot carry out his next move since none of his men has an unoccupied neighbor. If the game does not stop, then the outcome is a draw. We always assume that both players play optimal. Among other questions, we deal with the following ones: 1. 1. Is it true that, for all G and k, white cannot win the game? 2. 2. Does there exist a tree T and a positive integer k for which the outcome is a draw? Let τ(G) denote the covering number of G, i.e., τ(G) is the minimum number of vertices covering all edges of G. We prove that black wins the game if τ(G) ⩽ k. We use this result to show that white never wins the game if Gis bipartite, thus providing a partial answer to the first question. We answer the second question in the affirmative by constructing an infinite series of trees for which the outcome is a draw (for some k). Moreover, we present results on extremal problems arising in the context of the game. We also completely solve the cases when G is a path or a cycle. Further, we completely settle the case k ⩽ 2. In the proofs of our results, matchings and cycles in graphs play a predominant role.