A counterexample to the pseudo 2-factor isomorphic graph conjecture

Abstract

A graph $G$ is pseudo 2-factor isomorphic if the parity of the number of cycles in a 2-factor is the same for all 2-factors of $G$. Abreu et al. conjectured that $K_{3,3}$, the Heawood graph and the Pappus graph are the only essentially 4-edge-connected pseudo 2-factor isomorphic cubic bipartite graphs (Abreu et al., Journal of Combinatorial Theory, Series B, 2008, Conjecture 3.6). Using a computer search we show that this conjecture is false by constructing a counterexample with 30 vertices. We also show that this is the only counterexample up to at least 40 vertices. A graph $G$ is 2-factor hamiltonian if all 2-factors of $G$ are hamiltonian cycles. Funk et al. conjectured that every 2-factor hamiltonian cubic bipartite graph can be obtained from $K_{3,3}$ and the Heawood graph by applying repeated star products (Funk et al., Journal of Combinatorial Theory, Series B, 2003, Conjecture 3.2). We verify that this conjecture holds up to at least 40 vertices.Comment: 8 pages, added some extra information in Discrete Applied Mathematics (2015

Cubic graphs having only k-cycles in each 2-factor

“…Abreu et al [1] extended some results on 2-factor Hamiltonian graphs to the more general class of pseudo 2-factor isomorphic graphs, where a graph G is pseudo 2-factor isomorphic if the parity of the number of cycles in a 2-factor is the same for all 2-factors of G. They propose conjectures similar to Conjectures 9 and 10, for pseudo 2-factor isomorphic graphs. However, Goedgebeur [12] found a counterexample for those conjectures by a computer search, and also verified that Conjecture 10 is true up to 40 vertices.…”

A construction for a counterexample to the pseudo 2-factor isomorphic graph conjecture

“…In [7] Jan Goedgebeur computationally found a graph G on 30 vertices which is pseudo 2-factor isomorphic cubic and bipartite, essentially 4-edge-connected and cyclically 6edge-connected, thus refuting the above Conjecture 1.1 (cf. Figure 2).…”

“…All Levi graphs of other 15 3 configurations have automorphism group of order less than 4n (cf. [2], [3], [7]).…”