Abstract
The edge degree d ( e ) of the edge e = uv is defined as the number of neighbours of e, i.e., | N ( u ) ∪ N ( v ) | - 2 . Two edges are called remote if they are disjoint and there is no edge joining them. In this article, we prove that in a 2-connected graph G, if d ( e 1 ) + d ( e 2 ) > | V ( G ) | - 4 for any remote edges e 1 , e 2 , then all longest cycles C in G are dominating, i.e., G - V ( C ) is edgeless. This lower bound is best possible. As a corollary, it holds that if G is a 2-connected triangle-free graph with σ 2 ( G ) > | V ( G ) | / 2 , then all longest cycles are dominating.
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