Pairs of disjoint cycles

Serguei Norine, Robin Thomas, Hein van der Holst

Published 2017-11-12Version 1

Let $G=(V,E)$ be a finite undirected graph. Let $\mathbb{Z}\langle V\rangle$ denote free $\mathbb{Z}$-module generated by the vertices of $G$. Let $\mathbb{Z}\langle E\rangle$ denote the free $\mathbb{Z}$-module generated by the oriented edges of $G$. A $2$-cycle of $G$ is a bilinear form $d : \mathbb{Z}\langle E\rangle\times \mathbb{Z}\langle E\rangle\to \mathbb{Z}$ such for each edge $e$ of $G$, $d(e,\cdot)$ and $d(\cdot,e)$ are circulations, and $d(e,f) = 0$ whenever $e$ and $f$ have a common vertex. The $2$-cycles of a graph $G$ are in one-to-one correspondence with the homology classes in the second homology group of the deleted product of $G$. We show that each $2$-cycle is a linear combination of three special types of $2$-cycles: cycle-pair $2$-cycles, Kuratowski $2$-cycles, and quad $2$-cycles. Furthermore, we show that each skew-symmetric $2$-cycles is a linear combination of two special types of $2$-cycles: skew-symmetric cycle-pair $2$-cycles and skew-symmetric quad $2$-cycles.