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We study conditions under which an edge-coloured hypergraph has a particular substructure that contains more than the trivially guaranteed number of monochromatic edges. Our main result solves this problem for perfect matchings under minimum degree conditions. This answers recent questions of Gishboliner, Glock and Sgueglia, and of Balogh, Treglown and Z\'arate-Guer\'en.

We show that a $k$-uniform hypergraph on $n$ vertices has a spanning subgraph homeomorphic to the $(k - 1)$-dimensional sphere provided that $H$ has no isolated vertices and each set of $k - 1$ vertices supported by an edge is contained in at least $n/2 + o(n)$ edges. This gives a topological extension of Dirac's theorem and asymptotically confirms a conjecture of Georgakopoulos, Haslegrave, Montgomery, and Narayanan. Unlike typical results in the area, our proof does not rely on the Absorption Method, the Regularity Lemma or the Blow-up Lemma. Instead, we use a recently introduced framework that is based on covering the vertex set of the host graph with a family of complete blow-ups., Comment: 20 pages

We show that for all $k\geq 4$, $\varepsilon >0$, and $n$ sufficiently large, every $k$-uniform hypergraph on $n$ vertices in which each set of $k-3$ vertices is contained in at least $(5/8 + \varepsilon) \binom{n}{3}$ edges contains a tight Hamilton cycle. This is asymptotically best possible., Comment: 17 pages

We study conditions under which a given hypergraph is randomly robust Hamiltonian, which means that a random sparsification of the host graph contains a Hamilton cycle with high probability. Our main contribution provides nearly optimal results whenever the host graph is Hamilton connected in a locally robust sense, which translates to a typical induced subgraph of constant order containing Hamilton paths between any pair of suitable ends. The proofs are based on the recent breakthrough on Talagrand's conjecture, which reduces the problem to specifying a distribution on the desired guest structure in the (deterministic) host structure. We find such a distribution via a new argument that reduces the problem to the case of perfect matchings in a higher uniformity. As an application, we obtain asymptotically optimal results for perfect tilings in graphs and hypergraphs both in the minimum degree and uniformly dense setting. We also prove random robustness for powers of cycles under asymptotically optimal minimum degrees and degree sequences. We solve the problem for loose and tight Hamilton cycles in hypergraphs under a range of asymptotic minimum degree conditions. This includes in particular the case of $3$-uniform tight Hamilton cycles under vertex degree conditions. In all cases, our bounds on the sparseness are essentially best-possible., Comment: 28 pages

We study the emergence of loose Hamilton cycles in subgraphs of random hypergraphs. Our main result states that the minimum $d$-degree threshold for loose Hamiltonicity relative to the random $k$-uniform hypergraph $H_k(n,p)$ coincides with its dense analogue whenever $p \geq n^{- (k-1)/2+o(1)}$. The value of $p$ is approximately tight for $d>(k+1)/2$. This is particularly interesting because the dense threshold itself is not known beyond the cases when $d \geq k-2$., Comment: 33 pages, 3 figures

In the perfect tiling problem, we aim to cover the vertices of a hypergraph~$G$ with pairwise vertex-disjoint copies of a hypergraph $F$. There are three essentially necessary conditions for such a perfect tiling, which correspond to barriers in space, divisibility and covering. It is natural to ask in which situations these conditions are also asymptotically sufficient. Our main result confirms this for all hypergraph families that are hereditary in the sense of being approximately closed under taking typical induced subgraphs of constant order. Among others, this includes families parametrised by minimum degrees and quasirandomness, which have been studied extensively in this area. As an application, we recover and extend a series of well-known results for perfect tilings in hypergraphs and related settings involving vertex orderings and rainbow structures., Comment: 35 pages

Lehel conjectured in the 1970s that every red and blue edge-coloured complete graph can be partitioned into two monochromatic cycles. This was confirmed in 2010 by Bessy and Thomass\'e. However, the host graph $G$ does not have to be complete. It it suffices to require that $G$ has minimum degree at least $3n/4$, where $n$ is the order of $G$, as was shown recently by Letzter, confirming a conjecture of Balogh, Bar\'{a}t, Gerbner, Gy\'arf\'as and S\'ark\"ozy. This degree condition is asymptotically tight. Here we continue this line of research, by proving that for every red and blue edge-colouring of an $n$-vertex graph of minimum degree at least $2n/3 + o(n)$, there is a partition of the vertex set into three monochromatic cycles. This approximately verifies a conjecture of Pokrovskiy and is essentially tight.

We develop a method to study sufficient conditions for perfect mixed tilings. Our framework allows the embedding of bounded degree graphs $H$ with components of sublinear order. As a corollary, we recover and extend the work of K\"uhn and Osthus regarding sufficient minimum degree conditions for perfect $F$-tilings (for an arbitrary fixed graph $F$) by replacing the $F$-tiling with the aforementioned graphs $H$. Moreover, we obtain analogous results for degree sequences and in the setting of uniformly dense graphs. Finally, we asymptotically resolve a conjecture of Koml\'os in a strong sense., Comment: 32 pages + 2 pages appendix

We study structural conditions in dense graphs that guarantee the existence of vertex-spanning substructures such as Hamilton cycles. It is easy to see that every Hamiltonian graph is connected, has a perfect fractional matching and, excluding the bipartite case, contains an odd cycle. A simple consequence of the Robust Expander Theorem of K\"uhn, Osthus and Treglown tells us that any large enough graph that robustly satisfies these properties must already be Hamiltonian. Our main result generalises this phenomenon to powers of cycles and graphs of sublinear bandwidth subject to natural generalisations of connectivity, matchings and odd cycles. This answers a question of Ebsen, Maesaka, Reiher, Schacht and Sch\"ulke and solves the embedding problem that underlies multiple lines of research on sufficient conditions for spanning structures in dense graphs. As applications, we recover and establish Bandwidth Theorems in a variety of settings including Ore-type degree conditions, P\'osa-type degree conditions, deficiency-type conditions, locally dense and inseparable graphs, multipartite graphs as well as robust expanders.

In 1980, Akiyama, Exoo and Harary posited the Linear Arboricity Conjecture which states that any graph $G$ of maximum degree $\Delta$ can be decomposed into at most $\left\lceil \frac{\Delta}{2}\right\rceil$ linear forests. (A forest is linear if all of its components are paths.) In 1988, Alon proved the conjecture holds asymptotically. The current best bound is due to Ferber, Fox and Jain from 2020 who showed that $\frac{\Delta}{2}+ O(\Delta^{.661})$ suffices for large enough $\Delta$. Here, we show that $G$ admits a decomposition into at most $\frac{\Delta}{2}+ 3\sqrt{\Delta} \log^4 \Delta$ linear forests provided $\Delta$ is large enough. Moreover, our result also holds in the more general list setting, where edges have (possibly different) sets of permissible linear forests. Thus our bound also holds for the List Linear Arboricity Conjecture which was only recently shown to hold asymptotically by Kim and the second author. Indeed, our proof method ties together the Linear Arboricity Conjecture and the well-known List Colouring Conjecture; consequently, our error term for the Linear Arboricity Conjecture matches the best known error-term for the List Colouring Conjecture due to Molloy and Reed from 2000. This follows as we make two copies of every colour and then seek a proper edge colouring where we avoid bicoloured cycles between a colour and its copy; we achieve this via a clever modification of the nibble method.

The famous List Colouring Conjecture from the 1970s states that for every graph $G$ the chromatic index of $G$ is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds asymptotically. Our main result is a local generalization of Kahn's theorem. More precisely, we show that, for a graph $G$ with sufficiently large maximum degree $\Delta$ and minimum degree $\delta \geq \ln^{25} \Delta$, the following holds: for every assignment of lists of colours to the edges of $G$, such that $|L(e)| \geq (1+o(1)) \cdot \max\left\{\rm{deg}(u),\rm{deg}(v)\right\}$ for each edge $e=uv$, there is an $L$-edge-colouring of $G$. Furthermore, Kahn showed that the List Colouring Conjecture holds asymptotically for linear, $k$-uniform hypergraphs, and recently Molloy generalized Kahn's original result to correspondence colouring as well as its hypergraph generalization. We prove local versions of all of these generalizations by showing a weighted version that simultaneously implies all of our results., Comment: 25 pages, Accepted to JCTB

We develop a new framework to study minimum $d$-degree conditions in $k$-uniform hypergraphs, which guarantee the existence of a tight Hamilton cycle. Our main theoretical result deals with the typical absorption, path cover and connecting arguments for all $k$ and $d$ at once, and thus sheds light on the underlying structural problems. Building on this, we show that one can study minimum $d$-degree conditions of $k$-uniform tight Hamilton cycles by focusing on the inner structure of the neighbourhoods. This reduces the matter to an Erd\H{o}s--Gallai-type question for $(k-d)$-uniform hypergraphs, which is of independent interest. Once this framework is established, we can easily derive two new bounds. Firstly, we extend a classic result of R\"odl, Ruci\'nski and Szemer\'edi for $d=k-1$ by determining asymptotically best possible degree conditions for $d = k-2$ and all $k \ge 3$. This was proved independently by Polcyn, Reiher, R\"odl and Sch\"ulke. Secondly, we provide a general upper bound of $1-1/(2(k-d))$ for the tight Hamilton cycle $d$-degree threshold in $k$-uniform hypergraphs, thus narrowing the gap to the lower bound of $1-1/\sqrt{k-d}$ due to Han and Zhao., Comment: 59 pages, 4 figures. Accepted to JLMS

Given a family of graphs $\mathcal{F}$, we prove that the normalized edit distance of any given graph $\Gamma$ to being induced $\mathcal{F}$-free is estimable with a query complexity that depends only on the bounds of the Frieze--Kannan Regularity Lemma and on a Removal Lemma for $\mathcal{F}$., Comment: To appear in SIDMA

A classical result of Erd\H{o}s, Gy\'arf\'as and Pyber states that any $r$-edge-coloured complete graph has a partition into $O(r^2 \log r)$ monochromatic cycles. Here we determine the minimum degree threshold for this property. More precisely, we show that there exists a constant $c$ such that any $r$-edge-coloured graph on $n$ vertices with minimum degree at least $n/2 + c \cdot r \log n$ has a partition into $O(r^2)$ monochromatic cycles. We also provide constructions showing that the minimum degree condition and the number of cycles are essentially tight., Comment: 22 pages (26 including appendix)

We prove that in every $2$-colouring of the edges of $K_\mathbb{N}$ there exists a monochromatic infinite path $P$ such that $V(P)$ has upper density at least ${(12+\sqrt{8})}/{17} \approx 0.87226$ and further show that this is best possible. This settles a problem of Erd\H{o}s and Galvin., Comment: 16 pages, 2 figures

Erd\H{o}s, Gy\'arf\'as and Pyber showed that every $r$-edge-coloured complete graph $K_n$ can be covered by $25 r^2 \log r$ vertex-disjoint monochromatic cycles (independent of $n$). Here, we extend their result to the setting of binomial random graphs. That is, we show that if $p = p(n) = \Omega(n^{-1/(2r)})$, then with high probability any $r$-edge-coloured $G(n,p)$ can be covered by at most $1000 r^4 \log r $ vertex-disjoint monochromatic cycles. This answers a question of Kor\'andi, Mousset, Nenadov, \v{S}kori\'{c} and Sudakov., Comment: 16 pages, accepted in Combinatorics, Probability and Computing

There has been substantial interest in estimating the value of a graph parameter, i.e., of a real-valued function defined on the set of finite graphs, by querying a randomly sampled substructure whose size is independent of the size of the input. Graph parameters that may be successfully estimated in this way are said to be testable or estimable, and the sample complexity $q_z=q_z(\epsilon)$ of an estimable parameter $z$ is the size of a random sample of a graph $G$ required to ensure that the value of $z(G)$ may be estimated within an error of $\epsilon$ with probability at least 2/3. In this paper, for any fixed monotone graph property $\mathcal{P}=\mbox{Forb}(\mathcal{F})$, we study the sample complexity of estimating a bounded graph parameter $z_{\mathcal{P}}$ that, for an input graph $G$, counts the number of spanning subgraphs of $G$ that satisfy $\mathcal{P}$. To improve upon previous upper bounds on the sample complexity, we show that the vertex set of any graph that satisfies a monotone property $\mathcal{P}$ may be partitioned equitably into a constant number of classes in such a way that the cluster graph induced by the partition is not far from satisfying a natural weighted graph generalization of $\mathcal{P}$. Properties for which this holds are said to be recoverable, and the study of recoverable properties may be of independent interest.

We present a description of our method to process a set of autocollimator-based deflectometer 1-dimensional line-scans taken over a large optical surface and reconstruct them to a best-fit conic-section surface. The challenge with our task is that each line-scan is in a different (unknown) coordinate reference frame with respect to the other line-scans in the set. This problem arises due to the limited angular measurement range of the autocollimator used in the deflectometer and the need to measure over a greater range; this results in the optic under measurement being rotated (in pitch and roll) between each scan to bring the autocollimator back into measurement range and therefore each scan is taken in a different coordinate frame. We describe an approach using a 6N+2 dimension optimisation (where N is the number of scan lines taken across the mirror) that uses a gradient-based non-linear least squares fitting combined with a multi-start global search strategy to find the best-fit surface. Careful formulation of the problem is required to reduce numerical noise and allow the routine to converge on a solution of the required accuracy., Comment: 12 pages, 11 figures, Published in Applied Optics

We conjecture that any graph $G$ with treewidth~$k$ and maximum degree $\Delta(G)\geq k + \sqrt{k}$ satisfies $\chi'(G)=\Delta(G)$. In support of the conjecture we prove its fractional version. We also show that any graph $G$ with treewidth~$k\geq 4$ and maximum degree $2k-1$ satisfies $\chi'(G)=\Delta(G)$, improving an old result of Vizing., Comment: 14 pages, 3 figures, minor changes, accepted for publication in Electronic Journal of Combinatorics