Enclosings of Decompositions of Complete Multigraphs in 22-Edge-Connected rr-Factorizations

A decomposition of a multigraph $G$ is a partition of its edges into subgraphs $G(1), \ldots , G(k)$. It is called an $r$-factorization if every $G(i)$ is $r$-regular and spanning. If $G$ is a subgraph of $H$, a decomposition of $G$ is said to be enclosed in a decomposition of $H$ if, for every $1 \leq i \leq k$, $G(i)$ is a subgraph of $H(i)$. Feghali and Johnson gave necessary and sufficient conditions for a given decomposition of $\lambda K_n$ to be enclosed in some $2$-edge-connected $r$-factorization of $\mu K_{m}$ for some range of values for the parameters $n$, $m$, $\lambda$, $\mu$, $r$: $r=2$, $\mu>\lambda$ and either $m \geq 2n-1$, or $m=2n-2$ and $\mu = 2$ and $\lambda=1$, or $n=3$ and $m=4$. We generalize their result to every $r \geq 2$ and $m \geq 2n - 2$. We also give some sufficient conditions for enclosing a given decomposition of $\lambda K_n$ in some $2$-edge-connected $r$-factorization of $\mu K_{m}$ for every $r \geq 3$ and $m = (2 - C)n$, where $C$ is a constant that depends only on $r$, $\lambda$ and~$\mu$.Comment: 17 pages; fixed the proof of Theorem 1.4 and other minor change

A survey on constructive methods for the Oberwolfach problem and its variants

The generalized Oberwolfach problem asks for a decomposition of a graph $G$ into specified 2-regular spanning subgraphs $F_1,\ldots, F_k$, called factors. The classic Oberwolfach problem corresponds to the case when all of the factors are pairwise isomorphic, and $G$ is the complete graph of odd order or the complete graph of even order with the edges of a $1$-factor removed. When there are two possible factor types, it is called the Hamilton-Waterloo problem. In this paper we present a survey of constructive methods which have allowed recent progress in this area. Specifically, we consider blow-up type constructions, particularly as applied to the case when each factor consists of cycles of the same length. We consider the case when the factors are all bipartite (and hence consist of even cycles) and a method for using circulant graphs to find solutions. We also consider constructions which yield solutions with well-behaved automorphisms.Comment: To be published in the Fields Institute Communications book series. 23 pages, 2 figure

On uniformly resolvable (C4,K1,3)(C_4,K_{1,3})-designs

In this paper we consider the uniformly resolvable decompositions of the complete graph $K_v$ minus a 1-factor $(K_v − I)$ into subgraphs where each resolution class contains only blocks isomorphic to the same graph. We completely determine the spectrum for the case in which all the resolution classes consist of either 4-cycles or 3-stars

Deformation theory of deformed Hermitian Yang-Mills connections and deformed Donaldson-Thomas connections

A deformed Hermitian Yang-Mills (dHYM) connection and a deformed Donaldson-Thomas (dDT) connection are Hermitian connections on a Hermitian vector bundle $L$ over a K\"ahler manifold and a $G_2$-manifold, which are believed to correspond to a special Lagrangian and a (co)associative cycle via mirror symmetry, respectively. In this paper, when $L$ is a line bundle, introducing a new balanced Hermitian structure from the initial Hermitian structure and a dHYM connection and a new coclosed $G_2$-structure from the initial $G_2$-structure and a dDT connection, we show that their deformations are controlled by a subcomplex of the canonical complex introduced by Reyes Carri\'on. The expected dimension is given by the first Betti number of a base manifold for both cases. In the case of dHYM connections, we show that there are no obstructions for their deformations, and hence, the moduli space is always a smooth manifold. As an application of this, we give another proof of the triviality of the deformations of dHYM metrics proved by Jacob and Yau. In the case of dDT connections, we show that the moduli space is smooth if we perturb the initial $G_2$-structure generically.Comment: 56 pages, 1 figure, 4 tables, v2: a few typos are corrected, a reference is adde

Sugawara Construction and Casimir Operators for Krichever-Novikov Algebras

We show how to obtain from highest weight representations of Krichever-Novikov algebras of affine type (also called higher genus affine Kac-Moody algebras) representations of centrally extended Krichever-Novikov vector field algebras via the Sugawara construction. This generalizes classical results where one obtains representations of the Virasoro algebra. Relations between the weights of the corresponding representations are given and Casimir operators are constructed. In an appendix the Sugawara construction for the multi-point situation is done.Comment: Amstex 2.1, 39 page