Heavy subgraph pairs for traceability of block-chains

A graph is called traceable if it contains a Hamilton path, i.e., a pathcontaining all its vertices. Let G be a graph on n vertices. We say that aninduced subgraph of G is o−1-heavy if it contains two nonadjacent verticeswhich satisfy an Ore-type degree condition for traceability, i.e., with degreesum at least n−1 in G. A block-chain is a graph whose block graph is a path,i.e., it is either a P1, P2, or a 2-connected graph, or a graph with at least onecut vertex and exactly two end-blocks. Obviously, every traceable graph isa block-chain, but the reverse does not hold. In this paper we characterizeall the pairs of connected o−1-heavy graphs that guarantee traceability ofblock-chains. Our main result is a common extension of earlier work ondegree sum conditions, forbidden subgraph conditions and heavy subgraphconditions for traceability.