Weakly Split Graphs And Regular Cellulations Of The \(3\)-Sphere.

Abstract

We conjectured in \([3]\) that every biconnected cyclic graph is the one-dimensional skeleton of a regular cellulation of the \(3\)-sphere and proved it is true for planar and hamiltonian graphs. In this paper, we introduce the class of weakly split graphs and prove the conjecture is true for such class. Hamiltonian, split, complete \(k\)-partite, and matrogenic cyclic graphs are weakly split.