Circuit partitions and signed interlacement in 4-regular graphs

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 $\mathbb{R}$.

v1: 26 pages. v2: 26 pages; many small improvements. v3: 27 pages; minor corrections and changes in notation. v4: 28 pages, 3 figures; minor improvements. v5: 28 pages, 3 figures; minor improvements. v6: 28 pages, 3 figures; minor improvements. v7: 28 pages, 3 figures; updated references. v8: 25 pages, 4 figures. Further changes may be made before publication in Contrib. Discrete Math