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Richard Lang

Publications and source records attributed to Richard Lang.

31 records · Page 2Linked to original sources
arXiv2021

Minimum degree conditions for tight Hamilton cycles

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 Erdős--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 Rödl, Ruciński and Szemerédi 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, Rödl and Schülke. 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.

math.CO↗
arXiv2020

Minimum degree conditions for monochromatic cycle partitioning

A classical result of Erdős, Gyárfás 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.

math.CO↗
arXiv2020

Monochromatic cycle partitions in random graphs

Erdős, Gyárfás 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 $p = p(n) = Ω(n^{-1/(2r)})$, 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 Korándi, Mousset, Nenadov, Škorić and Sudakov.

math.CO↗
arXiv2019

Upper density of monochromatic infinite paths

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 Erdős and Galvin.

math.CO↗
arXiv2018

Chromatic index, treewidth and maximum degree

We conjecture that any graph $G$ with treewidth~$k$ and maximum degree $Δ(G)\geq k + \sqrt{k}$ satisfies $χ'(G)=Δ(G)$. 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 $χ'(G)=Δ(G)$, improving an old result of Vizing.

math.CO↗
arXiv2017

Estimating parameters associated with monotone properties

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 $q_z=q_z(ε)$ 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 $ε$ 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.

math.CO↗
arXiv2016

Local colourings and monochromatic partitions in complete bipartite graphs

We show that for any $2$-local colouring of the edges of the balanced complete bipartite graph $K_{n,n}$, its vertices can be covered with at most~$3$ disjoint monochromatic paths. And, we can cover almost all vertices of any complete or balanced complete bipartite $r$-locally coloured graph with $O(r^2)$ disjoint monochromatic cycles.\\ We also determine the $2$-local bipartite Ramsey number of a path almost exactly: Every $2$-local colouring of the edges of $K_{n,n}$ contains a monochromatic path on $n$ vertices.

math.CO↗
arXiv2016

Reconstruction of a conic-section surface from autocollimator-based deflectometric profilometry

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.

physics.optics↗
arXiv2013

List edge-colouring and total colouring in graphs of low treewidth

We prove that the list chromatic index of a graph of maximum degree $Δ$ and treewidth $\leq \sqrt{2Δ} -3$ is $Δ$; and that the total chromatic number of a graph of maximum degree $Δ$ and treewidth $\leq Δ/3 +1$ is $Δ+1$. This improves results by Meeks and Scott.

math.CO↗