On Theorems Equivalent with Kotzig's Result on Graphs with Unique 1-Factors

We show that several known theorems on graphs and digraphs are equivalent. The list of equivalent theorems include Kotzig’s result on graphs with unique 1-factors, a lemma by Seymour and Giles, theorems on alternating cycles in edge-colored graphs, and a theorem on semicycles in digraphs. We consider computational problems related to the quoted results; all these problems ask whether a given (di)graph contains a cycle satisfying certain properties which runs through p prescribed vertices. We show that all considered problems can be solved in polynomial time for p < 2 but are NP-complete for p ≥ 2.

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