The P5-saturation Game

Abstract

Let F, G and H be three graphs with G ⊆ H. We call G an F-saturated graph relative to H, if there is no copy of F in G but there is a copy of F in G + e for any e ∈ E(H) E(G). The F-saturation game on host graph H consists of two players, named Max and Min, who alternately add edges of H to G such that each chosen edge avoids creating a copy of F in G, and the players continue to choose edges until G becomes F-saturated relative to H. Max wishes to maximize the length of the game, while Min wishes to minimize the process. Let sat_g(F, H) (resp. sat′_g(F, H)) denote the number of edges chosen when Max (resp. when Min) starts the game and both players play optimally. In this article, we show that sat_g(P₅, K_n) = sat′_g(P₅, K_n) = n + 2 for n ≥ 15, and sat_g(P₅, K_m,n), sat′_g(P₅, K_m,n) lie in \(\left\{ {m + n - \left\lfloor {{{m - 2} \over 4}} \right\rfloor ,\,m + n - \left\lceil {{{m - 3} \over 4}} \right\rceil } \right\}\) if \(n \ge {5 \over 2}m\) and m ≥ 4, respectively.