On a base exchange game on bispanning graphs

Abstract

We consider the following maker-breaker game on a bispanning graph i.e. a graph that has a partition of the edge set E into two spanning trees E 1 and E 2 . Initially the edges of E 1 are purple and the edges of E 2 blue. Maker and breaker move alternately. In a move of the maker a blue edge is coloured purple. The breaker then has to recolour a different edge blue in such a way that the purple and the blue edges are spanning trees again. The goal of the maker is to exchange all colours, i.e. to make E 1 blue and E 2 purple. We prove that a sufficient but not necessary condition for the breaker to win is that the graph contains a K 4 . Furthermore we characterize the structure of a partition of a wheel into two spanning trees and show that the maker wins on wheels W n with n ≥ 4 and provide an example of a graph where, for some partitions, the maker wins, for some others, the breaker wins. We also describe an efficient algorithm for the recognition of bispanning graphs.