On 22-cycles of graphs

Let $G=(V,E)$ be a finite undirected graph. Orient the edges of $G$ in an arbitrary way. A $2$-cycle on $G$ is a function $d : E^2\to \mathbb{Z}$ such for each edge $e$, $d(e, \cdot)$ and $d(\cdot, e)$ are circulations on $G$, and $d(e, f) = 0$ whenever $e$ and $f$ have a common vertex. We show that each $2$-cycle is a sum of three special types of $2$-cycles: cycle-pair $2$-cycles, Kuratowski $2$-cycles, and quad $2$-cycles. In case that the graph is Kuratowski connected, we show that each $2$-cycle is a sum of cycle-pair $2$-cycles and at most one Kuratowski $2$-cycle. Furthermore, if $G$ is Kuratowski connected, we characterize when every Kuratowski $2$-cycle is a sum of cycle-pair $2$-cycles. A $2$-cycles $d$ on $G$ is skew-symmetric if $d(e,f) = -d(f,e)$ for all edges $e,f\in E$. We show that each $2$-cycle is a sum of two special types of skew-symmetric $2$-cycles: skew-symmetric cycle-pair $2$-cycles and skew-symmetric quad $2$-cycles. In case that the graph is Kuratowski connected, we show that each skew-symmetric $2$-cycle is a sum of skew-symmetric cycle-pair $2$-cycles. Similar results like this had previously been obtained by one of the authors for symmetric $2$-cycles. Symmetric $2$-cycles are $2$-cycles $d$ such that $d(e,f)=d(f,e)$ for all edges $e,f\in E$

On bilipschitz extensions in real Banach spaces

Suppose that $E$ and $E'$ denote real Banach spaces with dimension at least 2, that $D\not=E$ and $D'\not=E'$ are bounded domains with connected boundaries, that $f: D\to D'$ is an $M$-QH homeomorphism, and that $D'$ is uniform. The main aim of this paper is to prove that $f$ extends to a homeomorphism \bar \bar{D}\to \bar{D}' and $\bar{f}|\partial D$ is bilipschitz if and only if $f$ is bilipschitz in $\bar{D}$. The answer to some open problem of V\"ais\"al\"a is affirmative under an natural additional condition.Comment: arXiv admin note: substantial text overlap with arXiv:1105.468