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.