The Positive Minimum Degree Game on Sparse Graphs

Abstract

In this note we investigate a special form of degree games defined by D. Hefetz, M. Krivelevich, M. Stojaković and T. Szabó. Usually the board of a graph game is the edge set of $K_n$, the complete graph on $n$ vertices. Maker and Breaker alternately 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 first 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 Hefetz et al., we show, using the discharging method, that the sharp bound is around $10n/7$ for the positive minimum degree game.