Eulerian graphs and local complementation

Abstract

We define pure graphs, invertible graphs, and the notion of complementation of bicoloured graphs. The study of pure graphs is motivated by two conjectures about the transition systems of eulerian graphs and by the Cycle Double Cover Conjecture. We show how substitution rules can be used to determine when two complementation words produce the same graph. For bicoloured graphs, complementation words give way to complementation sets. The invertible graphs (bicoloured graphs whose vertex set is a complementation set) are shown to have the property that their inverse has the same automorphisms. The property of being invertible can also be defined for non-coloured graphs by endowing them with their natural colouring. It is proposed that a characterization of pure graphs would contribute to establish the truth of the Cycle Double Cover Conjecture. We show how pure graphs have essential factorizations into primitive pure graphs. The four primitive pure parity classes are presented.