Circuit Preserving Edge Maps II

Abstract

The results obtained in this paper grew from an attempt to generalize the main theorem of [1]. There it was shown that any circuit injection (a 1-1 onto edge map f such that if C is a circuit then f(C) is a circuit) from a 3-connected, not necessarily finite graph G onto a graph H is induced by a vertex isomorphism, where H is assumed to not have any isolated vertices. In the present article we examine the situation when the 1-1 condition is dropped (Chapter 1). An interesting result then is that the theorem remains true for finite (3-connected) graphs G but not for infinite G. In Chapter 2 we retain the 1-1 condition but allow the image of f to be first an arbitrary matroid and second a binary matroid. An interesting result then is the following. Let G be a graph of even order. Then the statement "no nontrivial map f:=>M exists, where M is a binary matroid" is equivalent to "G is Hamiltonian". If G is a graph of odd order, then the statement "no nontrivial map f:G=>M exists, where M is a binary matroid" is equivalent to "G is almost Hamiltonian", where we define a graph G of order n to be almost Hamiltonian if every subset of vertices of order n-1 is contained in some circuit of G. [1] J.H. Sanders and D. Sanders, Circuit preserving edge maps, J. Combin. Theory Ser. B 22 (1977),91-96.