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Characterizing forbidden pairs for hamiltonian properties. (English) Zbl 0879.05050
Given a family \(\mathcal F\) of graphs, a graph \(G\) is said to be \(\mathcal F\)-free if \(G\) contains no induced subgraph isomorphic to a member of \(\mathcal F\). Let \(P\) stand for any of the following graph-theoretic properties: traceable, hamiltonian, pancyclic, panconnected, cyclic extendable. The authors present a variety of theorems of the following general form:
Let \(R\) and \(S\) be connected graphs other than the path of length 2, and let the graph \(G\) have at least minimal connectivity necessary for property \(P\). Then, if \(|V(G)|\) is not too small (generally \(\geq 10\)), we have [\(G\) is \(\{R,S\}\)-free implies \(G\) has property \(P\)] if and only if [\(R=K_{1,3}\) and \(S\) is one of a short list of forbidden subgraphs].

05C45 Eulerian and Hamiltonian graphs
05C75 Structural characterization of families of graphs
05C38 Paths and cycles
05C40 Connectivity
Full Text: DOI
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