CANCELLING EULERIAN GRAPHS.

Abstract

If G is an (undirected) Eulerian graph, author labels the edges of G and defines 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 plus 1, negative if minus 1. 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. 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 grphs to the study of polynomial identities for skew-symmetric and symmetric matrices is also presented.