On the Space of 2-Linkages

TL;DR

This paper investigates the algebraic structure of 2-linkages in graphs, providing characterizations of when certain path span modules are proper subsets of a related module, under high connectivity conditions.

Contribution

It offers new characterizations of the space of 2-linkages in highly connected graphs, advancing understanding of their algebraic and combinatorial properties.

Findings

Characterizations of when $P(G; R, S)$ is a proper subset of $L(R, S)$

Results depend on the connectivity level of the graph

Provides conditions for the algebraic structure of 2-linkages

Abstract

Let $G=(V,E)$ be a finite undirected graph. If $P$ is an oriented path from $r_1\in V$ to $r_2\in V$, we define $\partial(P) = r_2-r_1$. If $R, S\subseteq V$, we denote by $P(G; R, S)$ the span of the set of all $\partial P\otimes \partial Q$ with $P$ and $Q$ disjoint oriented paths of $G$ connecting vertices in $R$ and $S$, respectively. By $L(R, S)$, we denote the submodule of $\mathbb{Z}\langle R\rangle\otimes\mathbb{Z}\langle S\rangle$ consisting all $\sum_{r\in R, s\in S} c(r,s)r\otimes s$ such that $c(r,r) = 0$ for all $r\in R\cap S$, $\sum_{r\in R} c(r, s) = 0$ for all $s\in S$, and $\sum_{s\in S} c(r, s) = 0$ for all $r\in R$. In this paper, we provide, when $G$ is sufficiently connected, characterizations when $P(G; R, S)$ is a proper subset of $L(R, S)$.