Edge-disjoint Hamilton cycles in 4-regular planar graphs

Abstract

The problem of determining which 4-regular graphs are decomposable into two edge-disjoint Hamilton cycles was first considered by Kotzig [4], who proved that a 3-regular graph is hamiltonian if and only if its edge graph (that is, line graph) has a hamiltonian decomposition. With the aid of this theorem, it is an easy matter to construct examples of 4-regular graphs which fail to have such a decomposition. Later, and presumably unaware of Kotzig's work (which was published in Slovak), Nash-Williams [7] conjectured that every 4-connected 4-regular graph admits a hamiltonian decomposition. Meredith [6] disproved this conjecture by constructing a nonhamiltonian 4-connected 4-regular graph. Gr/inbaum and Zaks [3] then proposed a weaker version of Nash-Williams' conjecture, restricted to planar graphs. (Recall that, by a theorem of Tutte [8], all 4-connected planar graphs are hamiltonian.) Counterexamples to this weaker conjecture were found by Gr/inbaum and Malkevitch [2] and also by Martin [5], who independently rediscovered Kotzig's theorem. The purpose of this note is to point out that Grinberg's necessary condition for a plane graph to be hamiltonian [1] can be used to derive a similar necessary condition for a 4-regular plane graph to admit a hamiltonian decomposition. To simplify the statement of Grinberg's theorem and its subsequent application, we make the following definition. If G is any plane graph with face set F, let g : 2 ~ ~ N be defined by