We consider the asymptotics of the difference between the empirical measures of the β-Hermite tridiagonal matrix and its minor. We prove that this difference has a deterministic limit and Gaussian fluctuations. Through a correspondence between measures and continual Young diagrams, this deterministic limit is identified with the Vershik–Kerov–Logan–Shepp curve. Moreover, the Gaussian fluctuations are identified with a sectional derivative of the Gaussian free field.

An n × n partial Latin square P is called α-dense if each row and column has at most αn non-empty cells and each symbol occurs at most αn times in P. An n × n array A where each cell contains a subset of {1,…, n} is a (βn, βn, βn)-array if each symbol occurs at most βn times in each row and column and each cell contains a set of size at most βn. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constants α, β > 0 such that, for every positive integer n, if P is an α-dense n × n partial Latin square, A is an n × n (βn, βn, βn)-array, and no cell of P contains a symbol that appears in the corresponding cell of A, then there is a completion of P that avoids A; that is, there is a Latin square L that agrees with P on every non-empty cell of P, and, for each i, j satisfying 1 ≤ i, j ≤ n, the symbol in position (i, j) in L does not appear in the corresponding cell of A.

Given complex numbers w1,…,wn, we define the weight w(X) of a set X of 0–1 vectors as the sum of $w_1^{x_1} \cdots w_n^{x_n}$ over all vectors (x1,…,xn) in X. We present an algorithm which, for a set X defined by a system of homogeneous linear equations with at most r variables per equation and at most c equations per variable, computes w(X) within relative error ∊ > 0 in (rc)O(lnn-ln∊) time provided $|w_j| \leq \beta (r \sqrt{c})^{-1}$ for an absolute constant β > 0 and all j = 1,…,n. A similar algorithm is constructed for computing the weight of a linear code over ${\mathbb F}_p$. Applications include counting weighted perfect matchings in hypergraphs, counting weighted graph homomorphisms, computing weight enumerators of linear codes with sparse code generating matrices, and computing the partition functions of the ferromagnetic Potts model at low temperatures and of the hard-core model at high fugacity on biregular bipartite graphs.

We study a restricted form of list colouring, for which every pair of lists that correspond to adjacent vertices may not share more than one colour. The optimal list size such that a proper list colouring is always possible given this restriction, we call separation choosability. We show for bipartite graphs that separation choosability increases with (the logarithm of) the minimum degree. This strengthens results of Molloy and Thron and, partially, of Alon. One attempt to drop the bipartiteness assumption precipitates a natural class of Ramsey-type questions, of independent interest. For example, does every triangle-free graph of minimum degree d contain a bipartite induced subgraph of minimum degree Ω(log d) as d→∞?

Let m(k) denote the maximum number of edges in a non-extendable, intersecting k-graph. Erdős and Lovász proved that m(k) ≤ kk. For k ≥ 625 we prove m(k) < kk・e−k1/4/6.

As a strengthening of Hadwiger’s conjecture, Gerards and Seymour conjectured that every graph with no odd Kt minor is (t − 1)-colourable. We prove two weaker variants of this conjecture. Firstly, we show that for each t ⩾ 2, every graph with no odd Kt minor has a partition of its vertex set into 6t − 9 sets V1, …, V6t−9 such that each Vi induces a subgraph of bounded maximum degree. Secondly, we prove that for each t ⩾ 2, every graph with no odd Kt minor has a partition of its vertex set into 10t −13 sets V1,…, V10t−13 such that each Vi induces a subgraph with components of bounded size. The second theorem improves a result of Kawarabayashi (2008), which states that the vertex set can be partitioned into 496t such sets.

For an edge-coloured graph G, the minimum colour degree of G means the minimum number of colours on edges which are incident to each vertex of G. We prove that if G is an edge-coloured graph with minimum colour degree at least 5, then V(G) can be partitioned into two parts such that each part induces a subgraph with minimum colour degree at least 2. We show this theorem by proving amuch stronger form. Moreover, we point out an important relationship between our theorem and Bermond and Thomassen’s conjecture in digraphs.

The strong chromatic number χs(G) of a graph G on n vertices is the least number r with the following property: after adding $r\lceil n/r\rceil-n$ isolated vertices to G and taking the union with any collection of spanning disjoint copies of Kr in the same vertex set, the resulting graph has a proper vertex colouring with r colours. We show that for every c > 0 and every graph G on n vertices with Δ(G) ≥ cn, χs(G) ≤ (2+o(1))Δ(G), which is asymptotically best possible.

We provide a deterministic algorithm that finds, in ɛ-O(1)n2 time, an ɛ-regular Frieze–Kannan partition of a graph on n vertices. The algorithm outputs an approximation of a given graph as a weighted sum of ɛ-O(1) many complete bipartite graphs.

As a corollary, we give a deterministic algorithm for estimating the number of copies of H in an n-vertex graph G up to an additive error of at most ɛnv(H), in time ɛ-OH(1)n2.

An (improper) graph colouring has defect d if each monochromatic subgraph has maximum degree at most d, and has clustering c if each monochromatic component has at most c vertices. This paper studies defective and clustered list-colourings for graphs with given maximum average degree. We prove that every graph with maximum average degree less than (2d+2)/(d+2)k is k-choosable with defect d. This improves upon a similar result by Havet and Sereni (J. Graph Theory, 2006). For clustered choosability of graphs with maximum average degree m, no (1-ɛ)m bound on the number of colours was previously known. The above result with d=1 solves this problem. It implies that every graph with maximum average degree m is $\lfloor{\frac{3}{4}m+1}\rfloor$-choosable with clustering 2. This extends a result of Kopreski and Yu (Discrete Math., 2017) to the setting of choosability. We then prove two results about clustered choosability that explore the trade-off between the number of colours and the clustering. In particular, we prove that every graph with maximum average degree m is $\lfloor{\frac{7}{10}m+1}\rfloor$-choosable with clustering 9, and is $\lfloor{\frac{2}{3}m+1}\rfloor$-choosable with clustering O(m). As an example, the later result implies that every biplanar graph is 8-choosable with bounded clustering. This is the best known result for the clustered version of the earth–moon problem. The results extend to the setting where we only consider the maximum average degree of subgraphs with at least some number of vertices. Several applications are presented.