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Abstract: A graph is called homogeneously traceable if for every vertex of , contains a Hamilton path starting from . For a graph , we say that is -free if contains no induced subgraph isomorphic to . For a family of graphs, is called -free if is -free for every . Determining families of graphs such that every -free graph has some graph property has been a popular research topic for several decades, especially for Hamiltonian properties, and more recently for properties related to the existence of graph factors. In this paper we give a complete characterization of all pairs of connected graphs such that every 2-connected -free graph is homogeneously traceable. [Copyright &y& Elsevier]

Abstract: Bollobás and Scott proved that if the weighted outdegree of every vertex of an edge-weighted digraph is at least 1, then the digraph contains a (directed) path of weight at least 1. In this note we characterize the extremal weighted digraphs with no heavy paths. Our result extends a corresponding theorem of Bondy and Fan on weighted graphs. We also give examples to show that a result of Bondy and Fan on the existence of heavy paths connecting two given vertices in a 2-connected weighted graph does not extend to 2-connected weighted digraphs. [Copyright &y& Elsevier]