A Size Upper Bound for Dominating Cycles

TL;DR

This paper establishes precise upper bounds on the size of 2-connected graphs that guarantee the longest cycle is dominating, extending previous results on Hamiltonian graphs with sharp bounds.

Contribution

It provides the exact size bounds for 2-connected graphs ensuring the longest cycle is dominating, generalizing earlier Hamiltonian conditions with sharp, proven limits.

Findings

For δ=2, if q ≤ 8, the longest cycle is dominating.

For δ ≥ 3, if q ≤ (3(δ-1)(δ+2)-1)/2, the longest cycle is dominating.

The bounds are proven to be sharp in all cases.

Abstract

Recently it was shown (by the author) that every graph of size $q$ (the number of edges) and minimum degree $\delta$ is hamiltonian if $q\le\delta^2+\delta-1$ (arXiv:1107.2201v1). In this paper we present the exact analog of this result for dominating cycles: if $G$ is a 2-connected graph with $q\le8$ if $\delta=2$ and $q\le (3(\delta-1)(\delta+2)-1)/2$ if $\delta\ge3$, then each longest cycle in $G$ is a dominating cycle. The result is sharp in all respects.