On the existence of uncountably many matroidal families

Abstract

A matroidal family is a set F ≠ ∅ of connected finite graphs such that for every finite graph G the edge-sets of those subgraphs of G which are isomorphic to some element of F are the circuits of a matroid on the edge-set of G. Simões-Pereira [5] shows the existence of four matroidal families and Andreae [1] shows the existence of a countably infinite series of matroidal families. In this paper we show that there exist uncountably many matroidal families. This is done by using an extension of Andreae's theorem, a construction theorem, and certain properties of regular graphs. Moreover we observe that all matroidal families so far known can be obtained in a unified way.