A note on the 2-Factor Hamiltonicity Conjecture

TL;DR

This work connects the 2-factor Hamiltonicity problem for cubic bipartite graphs to matching and Pfaffian graph theory, showing that the conjecture reduces to classifying cubic braces with all 2-factors Hamiltonian. It develops a tight-cut, brace-based reduction and leverages star-product and trisum constructions to transfer properties between graphs, ultimately proving that the Heawood graph is the only Pfaffian, cubic brace with all 2-factors Hamiltonian. A key intermediate result is that all Pfaffian braces aside from Heawood contain a cycle of length four, which underpins the girth-related conclusions and the reduction strategy. The findings illuminate the interplay between 2-factor Hamiltonicity and Pfaffian structure, with implications for brace decomposition, planar/nonplanar distinctions, and the use of star-product techniques in cubic bipartite graphs.

Abstract

The 2-factor Hamiltonicity Conjecture by Funk, Jackson, Labbate, and Sheehan [JCTB, 2003] asserts that all cubic, bipartite graphs in which all 2-factors are Hamiltonian cycles can be built using a simple operation starting from $K_{3,3}$ and the Heawood graph. We discuss the link between this conjecture and matching theory, in particular by showing that this conjecture is equivalent to the statement that the two exceptional graphs in the conjecture are the only cubic braces in which all 2-factors are Hamiltonian cycles, where braces are connected, bipartite graphs in which every matching of size at most two is contained in a perfect matching. In the context of matching theory this conjecture is especially noteworthy as $K_{3,3}$ and the Heawood graph are both strongly tied to the important class of Pfaffian graphs, with $K_{3,3}$ being the canonical non-Pfaffian graph and the Heawood graph being one of the most noteworthy Pfaffian graphs. Our main contribution is a proof that the Heawood graph is the only Pfaffian, cubic brace in which all 2-factors are Hamiltonian cycles. This is shown by establishing that, aside from the Heawood graph, all Pfaffian braces contain a cycle of length four, which may be of independent interest.

Figures (6)

• Figure 1: The graph depicted to the left is $K_{3,3}$. The drawing in the middle depicts the Heawood graph. On the very right a drawing of the cube is given.
• Figure 2: Two different star products of the same simple graph that exhibit unexpected behaviour.
• Figure 3: The graph on the right is the star product of two copies of the graph to the left.
• Figure 4: The graph on the right is the star product of two copies of the graph to the left.
• Figure 5: The graph to the right is the star product of the graph to the left and $K_4$. Since the graph to the right does not have a perfect matching, the principal 3-edge cut within it, marked with dashed lines, must be quasi-tight.

Theorems & Definitions (39)

• Conjecture 1.1: Funk, Jackson, Labbate, and Sheehan FunkJLS20032Factor
• Theorem 1.2: Little Little1975Characterization
• Theorem 1.3: Vazirani and Yannakakis VaziraniY1989Pfaffian
• Theorem 1.4: McCuaig McCuaig2004Polyas, Robertson, Seymour, and Thomas RobertsonST1999Permanents
• Theorem 1.5: Plummer Plummer1980$n$extendable
• Conjecture 1.6
• Theorem 1.7
• Corollary 1.8
• Conjecture 1.9
• Lemma 2.1: Funk, Jackson, Labbate, and Sheehan FunkJLS20032Factor