Endpoint extendable paths in dense graphs

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: • gree sum d ( u ) + d ( v ) ≥ n for any two nonadjacent vertices u and v ; free and ω ( G − S ) ≤ | S | for every cut set S of G ; ted, locally connected, and K 1 , 3 -free P 4 and K 1 , 3 denote a path of order 4 and a complete bipartite graph with 1 and 3 vertices in each color class, respectively.