Cancelling eulerian graphs

Abstract

If G is an (undirected) Eulerian graph, we label the edges of G and define the sign of an Eulerian path on G to be the sign of the associated permutation of the edges of the graph which is given by the Eulerian path. A path is positive if its sign is +1, negative if -1. One asks when a graph contains the same number of positive and negative Eulerian paths.

A vertex of a graph G is said to cancel if there are an equal number of positive and negative Eulerian paths which begin at the vertex. A graph is said to cancel if every vertex cancels. In fact, whether a graph cancels or not is independent of the choice of labels for the edges of the graph.

Properties of cancelling graphs are explored. Using results obtained by Swan for directed graphs, it can be shown that a graph with at least twice as many edges as vertices always cancels. The relevance of cancelling graphs to the study of polynomial identities for skew-symmetric and symmetric matrices will also be presented.