The positive minimum degree game on sparse graphs

In this note we investigate a special form of degree games de ned by D. Hefetz, M. Krivelevich, M. Stojaković and T. Szabó [4]. Usually the board of a graph game is the edge set of K_n, the complete graph on n vertices. Maker and Breaker alter-nately claim an edge, and Maker wins if his edges form a subgraph with prescribed properties; here a certain minimum degree. In the special form the board is no longer the whole edge set of K_n, Maker rst selects as few edges of K_n as possible in order to win, and our goal is to compute the necessary size of that board. Solving a question of [4], we show, using the discharging method, that the sharp bound is around 10n=7 for the positive minimum degree game.