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

Almost all optimally coloured complete graphs contain a rainbow Hamilton path

A subgraph $H$ of an edge-coloured graph is called rainbow if all of the edges of $H$ have different colours. In 1989, Andersen conjectured that every proper edge-colouring of $K_{n}$ admits a rainbow path of length $n-2$. We show that almost all optimal edge-colourings of $K_{n}$ admit both (i) a rainbow Hamilton path and (ii) a rainbow cycle using all of the colours. This result demonstrates that Andersen's Conjecture holds for almost all optimal edge-colourings of $K_{n}$ and answers a recent question of Ferber, Jain, and Sudakov. Our result also has applications to the existence of transversals in random symmetric Latin squares.Comment: 29 pages, 5 figure

On large-scale diagonalization techniques for the Anderson model of localization

We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation by Cullum and Willoughby with the shift-and-invert techniques in the implicitly restarted Lanczos method and in the Jacobi–Davidson method. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multilevel incomplete LDLT factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. We demonstrate the effectiveness and the numerical accuracy of these algorithms. Our numerical examples reveal that recent algebraic multilevel preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude

Infinitely many cyclic solutions to the Hamilton-Waterloo problem with odd length cycles

It is conjectured that for every pair $(\ell,m)$ of odd integers greater than 2 with $m \equiv 1\; \pmod{\ell}$, there exists a cyclic two-factorization of $K_{\ell m}$ having exactly $(m-1)/2$ factors of type $\ell^m$ and all the others of type $m^{\ell}$. The authors prove the conjecture in the affirmative when $\ell \equiv 1\; \pmod{4}$ and $m \geq \ell^2 -\ell + 1$.Comment: 31 page

On Hamilton cycles in graphs defined by intersecting set systems

In 1970 Lov\'asz conjectured that every connected vertex-transitive graph admits a Hamilton cycle, apart from five exceptional graphs. This conjecture has recently been settled for graphs defined by intersecting set systems, which feature prominently throughout combinatorics. In this expository article, we retrace these developments and give an overview of the many different ingredients in the proofs