Endpoint extendable paths in dense graphs

Abstract

A path in a graph is called extendable if it is a proper subpath of another path. A graph is locally connected if every neighborhood induces a connected subgraph. We show that, for each graph G of order n, there exists a threshold number s such that every path of order smaller than s is extendable and there exists a non-extendable path of order t for each t∈{s,…,n−1} if G satisfies any one of the following three conditions: •the degree sum d(u)+d(v)≥n for any two nonadjacent vertices u and v;•P4-free and ω(G−S)≤|S| for every cut set S of G;•connected, locally connected, and K1,3-free where P4 and K1,3 denote a path of order 4 and a complete bipartite graph with 1 and 3 vertices in each color class, respectively.