On Characterization of Petrie Partitionable Plane Graphs

Abstract

Given a plane graph \(G=(V,E)\), a Petrie tour of G is a tour P of G that alternately turns left and right at each step. A Petrie tour partition of G is a collection \(\mathcal{P}=\{P_1,\ldots ,P_q\}\) of Petrie tours so that each edge of G is in exactly one tour \(P_i \in \mathcal{P}\). A Petrie tour is called a Petrie cycle if all its vertices are distinct. A Petrie cycle partition of G is a collection \(\mathcal{C}=\{C_1,\ldots ,C_p\}\) of Petrie cycles so that each vertex of G is in exactly one cycle \(C_i \in \mathcal{C}\). In this paper, we characterize 3-regular (4-regular, resp.) plane graphs with Petrie cycle (tour, resp.) partitions. Given a 4-regular plane graph \(G=(V,E)\), a 3-regularization of G is a 3-regular plane graph \(G_3\) obtained from G by splitting every vertex \(v\in V\) into two degree-3 vertices. G is called Petrie partitionable if it has a 3-regularization that has a Petrie cycle partition. In this paper, we present an elegant characterization of Petrie partitionable graphs. The general version of this problem is motivated by a data compression method, tristrip, used in computer graphics.