Tutte paths and long cycles in circuit graphs

Thomassen proved that 4-connected planar graphs are Hamilton connected by showing that every 2-connected planar graph G contains a Tutte path P between any two given vertices, that is, every component of G−P has at most three neighbors on P. In this paper, we prove a quantitative version of this result for circuit graphs, a natural class of planar graphs which includes all 3-connected planar graphs, by further controlling the number of components in G−P. We also give an application of this result by providing a best possible bound for the circumference of essentially 4-connected planar graphs.