Non-Separating Cycles Through Prescirbed Vertices in Plane Graphs

Abstract

A graph G is k-connected if it has at least k + 1 vertices and remains connected after deleting k − 1 vertices of G. A plane graph is a graph G with an embedding on a plane R2 such that the interior of an edge contains no vertex and no point of any other edge. When G is a plane graph, we call the regions of R2\G the faces of G. A planar graph is a graph isomorphic to a plane graph. A plane triangulation is a plane graph in which each face is bounded by a 3-cycle. A graph G is k-linked if, for any given 2k vertices there are k disjoint paths joining each pair of them. For a given graph H, a graph G is H-linked if, for every injective map from the vertices of H to the vertices of G, G contains a subdivision of H. Let (K4 − e) be the graph obtained from K4 by removing one edge. A graph G is (K4 − e)-linked if, for every injective map from the vertices of K4 − e to the vertices of G, G contains a subdivision of K4 − e. A graph is said to be k-cyclable if given any set of k vertices there is a cycle that contains the k vertices. We say that a graph is k-ordered or Ck-linked if given any set of k vertices there is a cycle through the k vertices in any specified order. A graph is (K2 ∪ K3)-linked if for every set of two vertices and every set of three vertices there exists a path joining the two vertices and a cycle on the three vertices.

Seymour and Thomassen’s 2-linkage theorem characterizes all graphs which have two disjoint paths connecting any given two pair of vertices. Goddard proved that every 4-connected maximal planar graph is 4-ordered. Ellingham, Plummer, and Yu proved that any 4-connected planar triangulation is (K4 − e)-linked.

In this thesis we completely characterize the obstructions to (K2 ∪ K3)-linkage in 4-connected plane triangulations. From these obstructions we can see that no 4-connected nor 5-connected plane triangulation is (K2 ∪ K3)-linked. iii