Ramsey achievement games on graphs : algorithms and bounds

Abstract

In 1982, Harary introduced the concept of Ramsey achievement game on graphs. Given a graph F with no isolated vertices, consider the following game played on the complete graph \(K_n\) by two players Alice and Bob. First, Alice colors one of the edges of \(K_n\) blue, then Bob colors a different edge red, and so on. The first player who can complete the formation of F in his color is the winner. The minimum n for which Alice has a winning strategy is the achievement number of F, denoted by a(F). If we replace \(K_n\) in the game by the complete bipartite graph \(K_{n,n}\), we get the bipartite achievement number, denoted by \(\operatorname {ba}(F)\). In this paper, we correct \(\operatorname {ba}(mK_2)=m+1\) to m and disprove \(\operatorname {ba}(K_{1,m})=2m-2\) from Erickson and Harary (1983). We also find the exact values or upper and lower bounds of bipartite achievement numbers on matchings, stars, and double stars.