📝 SummarXiv

Abstract

This work is concerned with the study of the Game of Graph Nim -- a class of two-player combinatorial games -- on graphs with $4$ edges. To each edge of such a graph is assigned a positive-integer-valued edge-weight, and during each round of the game, the player whose turn it is to make a move selects a vertex, and removes a non-negative integer edge-weight from each of the edges incident on that vertex, such that the remaining edge-weight on each of these edges is a non-negative integer, and the total edge-weight removed during a round is strictly positive. The game continues for as long as the sum of the edge-weights remaining on all edges of the graph is strictly positive, and the player who plays the last round wins. An initial configuration of edge-weights is considered winning if the player who plays the first round wins the game, whereas it is defined as losing if the player who plays the second round wins. In this paper, we characterize, almost entirely, all winning and losing configurations for this game on all graphs with precisely $4$ edges each. Only one such graph defies our attempt to fully characterize the winning and losing configurations of edge-weights on its edges -- we are still able to provide a significant set of partial results pertaining to this graph.