A polynomial-time algorithm to determine (almost) Hamiltonicity of dense regular graphs
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We give a polynomial-time algorithm for detecting very long cycles in dense regular graphs. Specifically, we show that, given α∈ (0,1), there exists a c=c(α) such that the following holds: there is a polynomial-time algorithm that, given a D-regular graph G on n vertices with D≥α n, determines whether G contains a cycle on at least n - c vertices. The problem becomes NP-complete if we drop either the density or the regularity condition. The algorithm combines tools from extremal graph theory and spectral partitioning as well as some further algorithmic ingredients.
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