Pairwise disjoint perfect matchings in rr-edge-connected rr-regular graphs

Thomassen [Problem 1 in Factorizing regular graphs, J. Combin. Theory Ser. B, 141 (2020), 343-351] asked whether every $r$-edge-connected $r$-regular graph of even order has $r-2$ pairwise disjoint perfect matchings. We show that this is not the case if $r \equiv 2 \text{ mod } 4$. Together with a recent result of Mattiolo and Steffen [Highly edge-connected regular graphs without large factorizable subgraphs, J. Graph Theory, 99 (2022), 107-116] this solves Thomassen's problem for all even $r$. It turns out that our methods are limited to the even case of Thomassen's problem. We then prove some equivalences of statements on pairwise disjoint perfect matchings in highly edge-connected regular graphs, where the perfect matchings contain or avoid fixed sets of edges. Based on these results we relate statements on pairwise disjoint perfect matchings of 5-edge-connected 5-regular graphs to well-known conjectures for cubic graphs, such as the Fan-Raspaud Conjecture, the Berge-Fulkerson Conjecture and the $5$-Cycle Double Cover Conjecture.Comment: 24 page

When Do Measures on the Space of Connections Support the Triad Operators of Loop Quantum Gravity?

In this work we investigate the question, under what conditions Hilbert spaces that are induced by measures on the space of generalized connections carry a representation of certain non-Abelian analogues of the electric flux. We give the problem a precise mathematical formulation and start its investigation. For the technically simple case of U(1) as gauge group, we establish a number of "no-go theorems" asserting that for certain classes of measures, the flux operators can not be represented on the corresponding Hilbert spaces. The flux-observables we consider play an important role in loop quantum gravity since they can be defined without recourse to a background geometry, and they might also be of interest in the general context of quantization of non-Abelian gauge theories.Comment: LaTeX, 21 pages, 5 figures; v3: some typos and formulations corrected, some clarifications added, bibliography updated; article is now identical to published versio

Intersections of intrinsic submanifolds in the Heisenberg group

In the first Heisenberg group, we show that the intersection of two intrinsic submanifolds with linearly independent horizontal normals locally coincides with the image of an injective continuous curve. The key tool is a chain rule that relies on a recent result by Dafermos

The orientation morphism: from graph cocycles to deformations of Poisson structures

We recall the construction of the Kontsevich graph orientation morphism $\gamma \mapsto {\rm O\vec{r}}(\gamma)$ which maps cocycles $\gamma$ in the non-oriented graph complex to infinitesimal symmetries $\dot{\mathcal{P}} = {\rm O\vec{r}}(\gamma)(\mathcal{P})$ of Poisson bi-vectors on affine manifolds. We reveal in particular why there always exists a factorization of the Poisson cocycle condition $[\![\mathcal{P},{\rm O\vec{r}}(\gamma)(\mathcal{P})]\!] \doteq 0$ through the differential consequences of the Jacobi identity $[\![\mathcal{P},\mathcal{P}]\!]=0$ for Poisson bi-vectors $\mathcal{P}$. To illustrate the reasoning, we use the Kontsevich tetrahedral flow $\dot{\mathcal{P}} = {\rm O\vec{r}}(\gamma_3)(\mathcal{P})$, as well as the flow produced from the Kontsevich--Willwacher pentagon-wheel cocycle $\gamma_5$ and the new flow obtained from the heptagon-wheel cocycle $\gamma_7$ in the unoriented graph complex.Comment: 12 pages. Talk given by R.B. at Group32 (Jul 9--13, 2018; CVUT Prague, Czech Republic). Big formula in Appendix A retained from the (unpublished) Appendix in arXiv:1712.05259 [math-ph]. Signs corrected in v