Long cycles in graphs with large degree sums

Abstract

A number of results are established concerning long cycles in graphs with large degree sums. Let G be a graph on n vertices such that d( x)+ d( y)+ d( z)⩾ s for all triples of independent vertices x, y, z. Let c be the length of a longest cycle in G and α the cardinality of a maximum independent set of vertices. If G is 1-tough and s⩾ n, then every longest cycle in G is a dominating cycle and c⩾ min(n, n+ 1 3 s−α)⩾ min(n, 1 2 n+ 1 3 s)⩾ 5 6 n . If G is 2-connected and s⩾ n+2, then also c⩾ min(n, n+ 1 3 s-α) , generalizing a result of Bondy and one of Nash-Williams. Finally, if G is 2-tough and s⩾ n, then G is hamiltonian.