Circuit Preserving Edge Maps II

TL;DR

This paper explores the properties of circuit-preserving edge maps in graphs, extending previous theorems to infinite graphs and relating the existence of certain matroid maps to Hamiltonian properties.

Contribution

It generalizes circuit-preserving edge map theorems to infinite graphs and connects matroid mappings to Hamiltonian and almost Hamiltonian graph properties.

Findings

The main theorem holds for finite 3-connected graphs even when dropping injectivity.

Nontrivial matroid maps characterize Hamiltonian and almost Hamiltonian graphs.

The theorem does not extend to infinite graphs.

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.