Feedback game on [formula omitted]-chromatic Eulerian triangulations of surfaces

Abstract

A new impartial game on a connected graph was introduced, called a feedback game, which is a variant of generalized geography. In this paper, we study the feedback game on 3-chromatic Eulerian triangulations of surfaces. We prove that the winner of the game on every 3-chromatic Eulerian triangulation of a surface all of whose vertices have degree 0 modulo 4 is always fixed. Moreover, we also study the case of 3-chromatic Eulerian triangulations of surfaces which have at least two vertices whose degrees are 2 modulo 4. In addition, as a concrete class of such graphs, we consider the octahedral path, which is obtained from an octahedron by adding octahedra in the same face, and completely determine the winner of the game on those graphs.