Number of triangular packings of a marked graph on a projective plane

Abstract

A graph is called marked if its vertices differ one from the other by some label. As in [2], for a given marked graph G we denote by r(G, M 2) the number of all the pairwise different triangular packings (triangulations) Ti(G) of graph G on a connected, two-dimensional manifold (2-manifold) M2; moreover, two triangulations Til(G) and Ti2(G) with graph G will be

Irreducible triangulations of the Möbius band

“…Therefore, by Lemma 1, each irreducible triangulation of N 1 − D is obtainable either by removing a vertex from an irreducible triangulation in {P 1 , P 2 }, or by removing the pylonic vertex from a pylonic triangulation in {P 3 , P 4 , P 19 }. It is known [4,5,7] that Aut(P 1 ) acts transitively on the vertex set V (P 1 ), while under the action of Aut(P 2 ) the set V (P 2 ) breaks into two orbits as follows: orbit 1 = {1, 2, 3, 7}, orbit 2 = {4, 5, 6}. Therefore, all irreducible triangulations of N 1 − D are covered by the followings: M 1 = P 1 minus vertex 1 (subtructed with the incident edges and faces), M 2 = P 2 minus vertex 1, M 3 = P 2 minus vertex 4, M 4 = P 4 minus vertex 6 ′′ , M 5 = P 3 minus vertex 6 ′′ , M 6 = P 19 minus vertex 7 ′′ .…”