Regular bipartite graphs with all 2-factors isomorphic
The set BU\((k)\) is defined as the set of \(k\)-regular bipartite graphs with the property that all of their 2-factors are isomorphic. The main result of this paper is that if \(k\geq4\), then BU\((k)=\emptyset\). This result generalizes a result proved in the paper [\textit{M. Funk} et al., J. Comb. Theory, Ser. B 87, 138--144 (2003; Zbl 1045.05061)] which stated that if \(G\) is a \(k\)-regular bipartite graph where all of its 2-factors are Hamiltonian cycles, then \(k\leq3\). A graph whose 2-factors are all Hamiltonian cycles is said to be 2-factor Hamiltonian. The authors conjecture that a graph \(G\) is in BU\((k)\) if and only if each component of \(G\) is 2-factor Hamiltonian.
scientific article; zbMATH DE number 2104554