On the length of longest dominating cycles in graphs

Abstract

A cycle C in an undirected and simple graph G is dominating if G - C is edgeless. A graph G is called cycle-dominable if G contains a dominating cycle. There exists 1-tough graph in which no longest cycle is dominating. Moreover, the difference of the length of a longest cycle and of a longest dominating cycle in a 1-tough cycle-dominable graph may be made arbitrarily large. Some lower bounds for the length of dominating cycles in cycle-dominable graph are given. These results generalize and strengthen some well-known theorems of Jung and Fraisse (1989) and Bauer and Veldman et al. (1988).