On conjectures concerning the graph grabbing game

Abstract

We consider two conjectures made in regard to the graph grabbing game, played on a vertex weighted graph. Seacrest and Seacrest conjectured in 2012 that the first player can win the graph grabbing game on any even-order bipartite graph. Eoh and Choi conjectured a strengthening of this in 2019, namely that the first player can win on any graph with no induced corona product of an odd cycle and a point. We provide a family of counterexamples to the latter conjecture, and propose a weaker conjecture in its place. We also show that the above two conjectures are equivalent when the vertex weights are all $0$ or $1$.