Degree conditions and relative length of longest paths and cycles in graphs

Abstract

The relative length of a graph G, denoted by diff(G), is diff(G)=p(G)−c(G), where p(G) and c(G) denote the orders of a longest path and cycle, respectively. Let G be a 3-connected graph of order n. In this paper, we show that diff(G)≤1 if the maximum degree of any two nonadjacent vertices of G is more than (n+1)/3, and the lower bound is sharp. Moreover, if G is 2-connected graph satisfying the degree condition above, then diff(G)≤1 or G is in a special family of graphs.