Non-separating induced cycles in graphs

Abstract

In this paper we consider non-separating induced cycles in graphs. A basic result is that any 2-connected graph with at least six vertices and without such a cycle has at least four vertices of degree 2, and this is best possible. For any 3-connected graph G we prove that there exists a non-separating induced cycle C, such that all cycles in G-V(C) are contained in the same block of G-V(C). We apply our results in various directions. In particular, we obtain an extension of a conjecture of Hobbs (first proved by Jackson), and a new proof of Tutte's theorem on 3-connected graphs. Moreover, we show that any graph with minimum degree at least 3 contains a subdivision of K4 in which the three edges of a Hamiltonian path of the K4 are left undivided. This is an extension of a conjecture by Toft and implies an extension of a conjecture of Bollobás and Erdös (first proved by Larson) on the existence of an odd cycle with at least one diagonal. Finally, we obtain a result on the existence of a vertex joined by edges to three vertices of a cycle in a graph. This implies an extremal result conjectured by Bollobás and Erdös (first proved by Thomassen), as well as the conjecture of Toft that every 4-chromatic graph contains such a configuration.