On Petrie cycle and Petrie tour partitions of 3- and 4-regular plane graphs

Given a plane graph, 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 of Petrie tours so that each edge of G is in exactly one tour. A Petrie tour P is called a Petrie cycle if all its vertices are distinct. A Petrie cycle partition of G is a collection of Petrie cycles so that each vertex of G is in exactly one cycle. In this paper, we study the properties of 3-regular plane graphs that have Petrie cycle partitions and 4-regular plane multi-graphs that have Petrie tour partitions. Given a 4-regular plane multi-graph, a 3-regularization of G is a 3-regular plane graph obtained from G by splitting every vertex into two degree-3 vertices. G is called Petrie partitionable if it has a 3-regularization that has a Petrie cycle partition. The general version of this problem is motivated by a data compression method, tristrip, used in computer graphics. In this paper, we present a simple characterization of Petrie partitionable graphs and show that the problem of determining if G is Petrie partitionable is NP-complete.