Longest Paths and Longest Cycles in Graphs with Large Degree Sums

Abstract

Let p(G) and c(G) denote the number of vertices in a longest path and a longest cycle, respectively, of a finite, simple graph G. Define σ4(G)=min{d(x1)+d(x2)+ d(x3)+d(x4) | {x1,…,x4} is independent in G}. In this paper, the difference p(G)−c(G) is considered for 2-connected graphs G with σ4(G)≥|V(G)|+3. Among others, we show that p(G)−c(G)≤2 or every longest path in G is a dominating path.