Longest cycles and their chords

Abstract

Thomassen conjectured that any longest cycle of a 3-connected graph has a chord. In this paper, we will show that the conjecture is true for a planar graph if it is cubic or 6 2 4.

The 3‐connected graphs having a longest cycle containing only three contractible edges

“…Since then, there have been numerous other applications; in the context of this paper, Akira Saito (personal communication, 1989) has used contractible edge techniques to obtain the structure of a minimum counterexample (if one exists) to Thomassen's conjecture that every longest cycle in a 3-connected graph contains a chord. This result was first obtained by Zhang [18] using other methods.…”

Every longest circuit of a 3‐connected, K3,3‐minor free 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.…”

Chords in longest cycles

“…Then I realized that the method in [17] had a somewhat unexpected application, namely the chordconjecture restricted to cubic 3-connected graphs. (For planar cubic 3-connected graphs the conjecture was verified in [19].) Subsequently, the chord-conjecture was verified also E-mail address: ctho@dtu.dk.…”