3-Linkage on the Projective Plane

Abstract

This thesis will provide some brief background information in the fields of Graph Theory and Algebraic Topology, a short history of linkage problems, and then a new result for 3-linked graphs embedded in the projected plane. First we will prove some Algebraic Topology results, importantly that a punctured projective plane is homoemorphic to an open m\"{o}bius band. Then in the Graph Theory section, we will provide a proof of Menger's Theorem. Next, we will discuss the results near to that of this thesis, including the 2-linkage theorem and an extremal function of $k$-linkage. Finally, we will describe one structure which ensures three-linkage on the projected plane. Three-linkage is defined as follows: a graph $G$ is three-linked if for any three pairs $(s_1,t_1)$, $(s_2,t_2)$, $(s_3,t_3)$ of vertices in $G$, there exist vertex disjoint paths $P_1,P_2,P_3$ such that for $1\leq i \leq 3$, $P_i$ links $s_i$ to $t_i$. In this paper we will provide a classification of 5 conected graphs embedded in the projected plane with face-width at least 5. Namely, we will prove that if $G$ is a 5-connected graph embedded in the projective plane with face width at least 5, then $G$ is three-linked if and only if $G-s_3$ has has a specific structure.