Vertex-regular 11-factorizations in infinite graphs

The existence of $1$-factorizations of an infinite complete equipartite graph $K_m[n]$ (with $m$ parts of size $n$) admitting a vertex-regular automorphism group $G$ is known only when $n=1$ and $m$ is countable (that is, for countable complete graphs) and, in addition, $G$ is a finitely generated abelian group $G$ of order $m$. In this paper, we show that a vertex-regular $1$-factorization of $K_m[n]$ under the group $G$ exists if and only if $G$ has a subgroup $H$ of order $n$ whose index in $G$ is $m$. Furthermore, we provide a sufficient condition for an infinite Cayley graph to have a regular $1$-factorization. Finally, we construct 1-factorizations that contain a given subfactorization, both having a vertex-regular automorphism group

A complete solution to the infinite Oberwolfach problem

Let $F$ be a $2$-regular graph of order $v$. The Oberwolfach problem, $OP(F)$, asks for a $2$-factorization of the complete graph on $v$ vertices in which each $2$-factor is isomorphic to $F$. In this paper, we give a complete solution to the Oberwolfach problem over infinite complete graphs, proving the existence of solutions that are regular under the action of a given involution free group $G$. We will also consider the same problem in the more general contest of graphs $F$ that are spanning subgraphs of an infinite complete graph $\mathbb{K}$ and we provide a solution when $F$ is locally finite. Moreover, we characterize the infinite subgraphs $L$ of $F$ such that there exists a solution to $OP(F)$ containing a solution to $OP(L)$

Ten Conferences WORDS: Open Problems and Conjectures

In connection to the development of the field of Combinatorics on Words, we present a list of open problems and conjectures that were stated during the ten last meetings WORDS. We wish to continually update the present document by adding informations concerning advances in problems solving

Unique factorization in perturbative QFT

We discuss factorization of the Dyson--Schwinger equations using the Lie- and Hopf algebra of graphs. The structure of those equations allows to introduce a commutative associative product on 1PI graphs. In scalar field theories, this product vanishes if and only if one of the factors vanishes. Gauge theories are more subtle: integrality relates to gauge symmetries.Comment: 5pages, Talk given at "RadCor 2002 - Loops and Legs 2002", Kloster Banz, Germany, Sep 8-13, 200

Matrix factorizations and link homology

For each positive integer n the HOMFLY polynomial of links specializes to a one-variable polynomial that can be recovered from the representation theory of quantum sl(n). For each such n we build a doubly-graded homology theory of links with this polynomial as the Euler characteristic. The core of our construction utilizes the theory of matrix factorizations, which provide a linear algebra description of maximal Cohen-Macaulay modules on isolated hypersurface singularities.Comment: 108 pages, 61 figures, latex, ep