Graphs with multi-$4$-cycles and the Barnette's conjecture

Abstract

Let H denote the family of all graphs with multi-4-cycles and suppose that G ∈ H. Then, G is a bipartite graph with a vertex bipartition {V α , V β }. We prove that for every vertex v ∈ V β and for every 2-colouringLet now G be a simple even plane triangulation with a vertex 3-partitionthen, for any edge chosen on a face coloured 3 and of size at least 6 in G * , there exists a Hamilton cycle of G * which avoids this edge. Moreover, if every component of H is 2-connected, then there exists a Hamilton cycle of G * such that for every face coloured 3 it avoids every second edge of this face or it avoids at most two edges of this face.