Every longest circuit of a 3-connected, K3, 3-minor free graph has a chord

Carsten Thomassen conjectured that every longest circuit in a 3-connected graph has a chord. We prove the conjecture for graphs having no K3,3 minor, and consequently for planar graphs. Carsten Thomassen made the following conjecture [1, 7]: Conjecture 1 (Thomassen) Every longest circuit of a 3-connected graph has a chord. That conjecture has been proved for planar graphs with minimum degree at least four [9], cubic graphs [8] and graphs embeddable in several surfaces [4, 5, 6]. In this paper, we prove it for planar graphs in general. In fact, our result concerns a class of graphs which contains the planar graphs. Let us denote by K3,3 the complete bipartite graph drawn in Figure 1. A minor of a graph G is a graph H which can be obtained from G by a sequence of vertex deletions, edge deletions and edge contractions. For more details about graph minors or the classical notations of graph theory used through the paper, the reader can refer to any general book on graph theory, for instance [2, 3]. The main result of this paper is the following: Theorem 2 Let G be a 3-connected graph with no K3,3 minor. Every longest circuit of G has a chord. Denoting by K the complete graph on five vertices, Kuratowski’s theorem [3] states that a graph is planar if and only if it contains neither K nor K3,3 as a minor. Therefore, Theorem 2 immediatly yields: Corollary 3 Every longest circuit in a planar 3-connected graph has a chord.