Circuit partitions and signed interlacement in 4-regular graphs

TL;DR

This paper explores the relationship between circuit partitions, interlacement matrices, and touch-graphs in 4-regular graphs, revealing new algebraic connections over GF(2) and real numbers.

Contribution

It introduces a novel connection between interlacement matrices of Euler systems and the cycle space of associated touch-graphs in 4-regular graphs.

Findings

Established links between interlacement matrices and cycle spaces.

Extended analysis over GF(2) and real numbers.

Provided new algebraic tools for graph circuit analysis.

Abstract

Let $F$ be a 4-regular graph. Each circuit partition $P$ of $F$ has a corresponding touch-graph $Tch(P)$; the circuits in $P$ correspond to vertices of $Tch(P)$, and the vertices of $F$ correspond to edges of $Tch(P)$. We discuss the connection between modified versions of the interlacement matrix of an Euler system of $F$ and the cycle space of $Tch(P)$, over $GF(2)$ and $\mathbb{R}$.