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.

• Type: Journal
• Year: 2004
• Where: ARSCOM
• Authors: Stefan Szeider
• Keywords: ARSCOM 2004, Equivalent Theorems, Theorems, Unique 1-factors