Cycle type in Hall-Paige: A proof of the Friedlander-Gordon-Tannenbaum conjecture

An orthomorphism of a finite group $G$ is a bijection $\phi\colon G\to G$ such that $g\mapsto g^{-1}\phi(g)$ is also a bijection. In 1981, Friedlander, Gordon, and Tannenbaum conjectured that when $G$ is abelian, for any $k\geq 2$ dividing $|G|-1$, there exists an orthomorphism of $G$ fixing the identity and permuting the remaining elements as products of disjoint $k$-cycles. We prove this conjecture for all sufficiently large groups.Comment: 34 page

Unavoidable patterns in locally balanced colourings

Which patterns must a two-colouring of $K_n$ contain if each vertex has at least $\varepsilon n$ red and $\varepsilon n$ blue neighbours? In this paper, we investigate this question and its multicolour variant. For instance, we show that any such graph contains a $t$-blow-up of an \textit{alternating 4-cycle} with $t = \Omega(\log n)$.Comment: Improved expositio

A random Hall-Paige conjecture

A complete mapping of a group $G$ is a bijection $\phi\colon G\to G$ such that $x\mapsto x\phi(x)$ is also bijective. Hall and Paige conjectured in 1955 that a finite group $G$ has a complete mapping whenever $\prod_{x\in G} x$ is the identity in the abelianization of $G$. This was confirmed in 2009 by Wilcox, Evans, and Bray with a proof using the classification of finite simple groups. In this paper, we give a combinatorial proof of a far-reaching generalisation of the Hall-Paige conjecture for large groups. We show that for random-like and equal-sized subsets $A,B,C$ of a group $G$, there exists a bijection $\phi\colon A\to B$ such that $x\mapsto x\phi(x)$ is a bijection from $A$ to $C$ whenever $\prod_{a\in A} a \prod_{b\in B} b=\prod_{c\in C} c$ in the abelianization of $G$. Using this result, we settle the following conjectures for sufficiently large groups. (1) We confirm in a strong form a conjecture of Snevily by characterising large subsquares of multiplication tables of finite groups that admit transversals. Previously, this characterisation was known only for abelian groups of odd order. (2) We characterise the abelian groups that can be partitioned into zero-sum sets of specific sizes, solving a problem of Tannenbaum, and confirming a conjecture of Cichacz. (3) We characterise harmonious groups, that is, groups with an ordering in which the product of each consecutive pair of elements is distinct, solving a problem of Evans. (4) We characterise the groups with which any path can be assigned a cordial labelling. In the case of abelian groups, this confirms a conjecture of Patrias and Pechenik

Transversal factors and spanning trees

Given a collection of graphs $\mathbf{G}=(G_1, \ldots, G_m)$ with the same vertex set, an $m$-edge graph $H\subset \cup_{i\in [m]}G_i$ is a transversal if there is a bijection $\phi:E(H)\to [m]$ such that $e\in E(G_{\phi(e)})$ for each $e\in E(H)$. We give asymptotically-tight minimum degree conditions for a graph collection on an $n$-vertex set to have a transversal which is a copy of a graph $H$, when $H$ is an $n$-vertex graph which is an $F$-factor or a tree with maximum degree $o(n/\log n)$.Comment: 21 page

Optimal spread for spanning subgraphs of Dirac hypergraphs

Let $G$ and $H$ be hypergraphs on $n$ vertices, and suppose $H$ has large enough minimum degree to necessarily contain a copy of $G$ as a subgraph. We give a general method to randomly embed $G$ into $H$ with good "spread". More precisely, for a wide class of $G$, we find a randomised embedding $f\colon G\hookrightarrow H$ with the following property: for every $s$, for any partial embedding $f'$ of $s$ vertices of $G$ into $H$, the probability that $f$ extends $f'$ is at most $O(1/n)^s$. This is a common generalisation of several streams of research surrounding the classical Dirac-type problem. For example, setting $s=n$, we obtain an asymptotically tight lower bound on the number of embeddings of $G$ into $H$. This recovers and extends recent results of Glock, Gould, Joos, K\"uhn, and Osthus and of Montgomery and Pavez-Sign\'e regarding enumerating Hamilton cycles in Dirac hypergraphs. Moreover, using the recent developments surrounding the Kahn--Kalai conjecture, this result implies that many Dirac-type results hold robustly, meaning $G$ still embeds into $H$ after a random sparsification of its edge set. This allows us to recover a recent result of Kang, Kelly, K\"uhn, Osthus, and Pfenninger and of Pham, Sah, Sawhney, and Simkin for perfect matchings, and obtain novel results for Hamilton cycles and factors in Dirac hypergraphs. Notably, our randomised embedding algorithm is self-contained and does not require Szemer\'edi's regularity lemma or iterative absorption.Comment: 26 page

Splitting matchings and the Ryser-Brualdi-Stein conjecture for multisets

We study multigraphs whose edge-sets are the union of three perfect matchings, $M_1$, $M_2$, and $M_3$. Given such a graph $G$ and any $a_1,a_2,a_3\in \mathbb{N}$ with $a_1+a_2+a_3\leq n-2$, we show there exists a matching $M$ of $G$ with $|M\cap M_i|=a_i$ for each $i\in \{1,2,3\}$. The bound $n-2$ in the theorem is best possible in general. We conjecture however that if $G$ is bipartite, the same result holds with $n-2$ replaced by $n-1$. We give a construction that shows such a result would be tight. We also make a conjecture generalising the Ryser-Brualdi-Stein conjecture with colour multiplicities