On Tutte cycles containing three prescribed edges
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 2connected 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 4connected plane graphs that also contain three prescribed edges of a facial cycle.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 DOI:
 10.48550/arXiv.2106.09617
 arXiv:
 arXiv:2106.09617
 Bibcode:
 2021arXiv210609617W
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 14 pages, 1 figure