Characterization of graphs without even F-orientations

A graph G is 1-extendable if every edge belongs to at least one 1-factor of G .Let G be a graph with a 1 factor F . Then an even (odd) F -orientation of G is an orientation in which each F -alternating cycle has exactly an even (odd) number of edges directed in the same fixed direction around the cycle. If a graph G admits an odd F -orientation for some 1-factor F then its admits an odd F ′-orientation for all 1-factors F ′. Such graphs are called Pfaffian and have been widely studied. A similar statement is not true for even F -orientations, of which little is known. The purpose of this paper is to achieve helpful results about even F-orientations. In particular, we examine the structure of 1-extendable graphs G which have no even F -orientation for a fixed 1–factor F of G . Such graphs contain a special subgraph in a family that we will call generalized Wagner graphs. We give a complete characterization of generalized Wagner graphs without even F orientations in the case of connectivity at least four and in the case of k -regular graphs for k ≥ 3.