Scattering number and extremal non-Hamiltonian graphs
Discrete Mathematics
4-connected projective-planar graphs are Hamiltonian
Journal of Combinatorial Theory Series B
On Hamilton cycles in certain planar graphs
Journal of Graph Theory
Journal of Graph Theory
Five-connected toroidal graphs are Hamiltonian
Journal of Combinatorial Theory Series B
Measuring the vulnerability for classes of intersection graphs
Discrete Applied Mathematics
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Hamilton paths in toroidal graphs
Journal of Combinatorial Theory Series B
Thomassen's conjecture implies polynomiality of 1-Hamilton-connectedness in line graphs
Journal of Graph Theory
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A graph G is called 2-edge-Hamiltonian-connected if for any X@?{x"1x"2:x"1,x"2@?V(G)} with 1@?|X|@?2, G@?X has a Hamiltonian cycle containing all edges in X, where G@?X is the graph obtained from G by adding all edges in X. In this paper, we show that every 4-connected plane graph is 2-edge-Hamiltonian-connected. This result is best possible in many senses and an extension of several known results on Hamiltonicity of 4-connected plane graphs, for example, Tutte's result saying that every 4-connected plane graph is Hamiltonian, and Thomassen's result saying that every 4-connected plane graph is Hamiltonian-connected. We also show that although the problem of deciding whether a given graph is 2-edge-Hamiltonian-connected is NP-complete, there exists a polynomial time algorithm to solve the problem if we restrict the input to plane graphs.