On Tutte cycles containing three prescribed edges

TL;DR

This paper investigates Tutte cycles in plane graphs, providing bounds on the number of components after deletion and demonstrating the existence of long cycles containing three specified edges, advancing understanding of Hamiltonicity.

Contribution

It offers a quantitative version of Tutte cycle existence with bounds on components, and shows how to find long cycles with prescribed edges in 4-connected plane graphs.

Findings

Bound on the number of components after deleting a Tutte cycle.

Existence of long cycles containing three prescribed edges.

Extension of previous results to a quantitative framework.

Abstract

A cycle $C$ in a graph $G$ is called a Tutte cycle if, after deleting $C$ from $G$, each component has at most three neighbors on $C$. Tutte cycles play an important role in the study of Hamiltonicity of planar graphs. Thomas and Yu and independently Sanders proved the existence of Tutte cycles containining three specified edges of a facial cycle in a 2-connected plane graph. We prove a quantitative version of this result, bounding the number of components of the graph obtained by deleting a Tutte cycle. As a corollary, we can find long cycles in essentially 4-connected plane graphs that also contain three prescribed edges of a facial cycle.