What simple condition on a graph $G$ will ensure that $G\,{\succ}\,K_t$? As usual, $G\,{\succ}\,K_t$ means that $K_t$ is a minor of the graph $G$ (in other words, $G$ has vertex disjoint connected subgraphs $W_1,\ldots,W_t$ and at least one edge between $W_i$ and $W_j$, $1\,{\le}\, i<j\,{\le}\, t$).

In this note it is shown that resistance and resistance dimension are rough isometry invariant.

Every general graph with degrees 2k and 2k − 2, k ≥ 3, with zero or at least two vertices of degree 2k − 2 in each component, has a k-edge-colouring such that each monochromatic subgraph has degree 1 or 2 at every vertex.

In particular, if T is a triangle in a 6-regular general graph, there exists a 2-factorization of G such that each factor uses an edge in T if and only if T is non-separating.

This paper deals with the partition function of the Ising model from statistical mechanics, which is used to study phase transitions in physical systems. A special case of interest is that of the Ising model with constant energies and external field. One may consider such an Ising system as a simple graph together with vertex and edge weights. When these weights are considered indeterminates, the partition function for the constant case is a trivariate polynomial Z(G;x,y,z). This polynomial was studied with respect to its approximability by Goldberg, Jerrum and Paterson. Z(G;x,y,z) generalizes a bivariate polynomial Z(G;t,y), which was studied in by Andrén and Markström.

We consider the complexity of Z(Gt,y) and Z(G;x,y,z) in comparison to that of the Tutte polynomial, which is well known to be closely related to the Potts model in the absence of an external field. We show that Z(G;x,y,z) is #P-hard to evaluate at all points in 3, except those in an exceptional set of low dimension, even when restricted to simple graphs which are bipartite and planar. A counting version of the Exponential Time Hypothesis, #ETH, was introduced by Dell, Husfeldt and Wahlén in order to study the complexity of the Tutte polynomial. In analogy to their results, we give under #ETH a dichotomy theorem stating that evaluations of Z(G;t,y) either take exponential time in the number of vertices of G to compute, or can be done in polynomial time. Finally, we give an algorithm for computing Z(G;x,y,z) in polynomial time on graphs of bounded clique-width, which is not known in the case of the Tutte polynomial.

We consider the problem of sampling ‘unlabelled structures’, i.e., sampling combinatorial structures modulo a group of symmetries. The main tool which has been used for this sampling problem is Burnside’s lemma. In situations where a significant proportion of the structures have no nontrivial symmetries, it is already fairly well understood how to apply this tool. More generally, it is possible to obtain nearly uniform samples by simulating a Markov chain that we call the Burnside process: this is a random walk on a bipartite graph which essentially implements Burnside’s lemma. For this approach to be feasible, the Markov chain ought to be ‘rapidly mixing’, i.e., converge rapidly to equilibrium. The Burnside process was known to be rapidly mixing for some special groups, and it has even been implemented in some computational group theory algorithms. In this paper, we show that the Burnside process is not rapidly mixing in general. In particular, we construct an infinite family of permutation groups for which we show that the mixing time is exponential in the degree of the group.

In a recent paper, Baxter and Zeilberger showed that the two most important Mahonian statistics, the inversion number and the major index, are asymptotically independently normally distributed on permutations. In another recent paper, Canfield, Janson and Zeilberger proved the result, already known to statisticians, that the Mahonian distribution is asymptotically normal on words. This leaves one question unanswered: What, asymptotically, is the joint distribution of the inversion number and the major index on words? We answer this question by establishing convergence to a bivariate normal distribution.

An equitable colouring of a graph G is a vertex colouring where no two adjacent vertices are coloured the same and, additionally, the colour class sizes differ by at most 1. The equitable chromatic number χ=(G) is the minimum number of colours required for this. We study the equitable chromatic number of the dense random graph ${\mathcal{G}(n,m)}$ where $m = \left\lfloor {p\left( \matrix{ n \cr 2 \cr} \right)} \right\rfloor $ and 0 < p < 0.86 is constant. It is a well-known question of Bollobás [3] whether for p = 1/2 there is a function f(n) → ∞ such that, for any sequence of intervals of length f(n), the normal chromatic number of ${\mathcal{G}(n,m)}$ lies outside the intervals with probability at least 1/2 if n is large enough. Bollobás proposes that this is likely to hold for f(n) = log n. We show that for the equitable chromatic number, the answer to the analogous question is negative. In fact, there is a subsequence ${({n_j})_j}_{ \in {\mathbb {N}}}$ of the integers where $\chi_=({\mathcal{G}(n_j,m_j)})$ is concentrated on exactly one explicitly known value. This constitutes surprisingly narrow concentration since in this range the equitable chromatic number, like the normal chromatic number, is rather large in absolute value, namely asymptotically equal to n/(2logbn) where b = 1/(1 − p).

Let G be a graph on n vertices and with maximum degree Δ, and let k be an integer. The k-recolouring graph of G is the graph whose vertices are k-colourings of G and where two k-colourings are adjacent if they differ at exactly one vertex. It is well known that the k-recolouring graph is connected for $k\geq \Delta+2$ . Feghali, Johnson and Paulusma (J. Graph Theory83 (2016) 340–358) showed that the (Δ + 1)-recolouring graph is composed by a unique connected component of size at least 2 and (possibly many) isolated vertices.

In this paper, we study the proportion of isolated vertices (also called frozen colourings) in the (Δ + 1)-recolouring graph. Our first contribution is to show that if G is connected, the proportion of frozen colourings of G is exponentially smaller in n than the total number of colourings. This motivates the use of the Glauber dynamics to approximate the number of (Δ + 1)-colourings of a graph. In contrast to the conjectured mixing time of O(nlog n) for $k\geq \Delta+2$ colours, we show that the mixing time of the Glauber dynamics for (Δ + 1)-colourings restricted to non-frozen colourings can be Ω(n2). Finally, we prove some results about the existence of graphs with large girth and frozen colourings, and study frozen colourings in random regular graphs.

Alweiss, Lovett, Wu, and Zhang introduced $q$-spread hypergraphs in their breakthrough work regarding the sunflower conjecture, and since then $q$-spread hypergraphs have been used to give short proofs of several outstanding problems in probabilistic combinatorics. A variant of $q$-spread hypergraphs was implicitly used by Kahn, Narayanan, and Park to determine the threshold for when a square of a Hamiltonian cycle appears in the random graph $G_{n,p}$. In this paper, we give a common generalization of the original notion of $q$-spread hypergraphs and the variant used by Kahn, Narayanan, and Park.

The authors would like to rectify a mistake made on the second page, line 50 of their article. The text below explains the changes required.

The constant 0.628 on line 50 of page 2 should be 0.638.

In this note, I will present some new problems, and also some old ones which, in my opinion, have been undeservedly forgotten or neglected.

Consider the set of finite words on a totally ordered alphabet with two letters. We prove that the distribution of the length of the standard right factor of a random Lyndon word with length n, divided by n, converges to when n goes to infinity. The convergence of all moments follows. This paper thus completes the results of [2], in which the limit of the first moment is given.

The authors would like to rectify a mistake made in Theorem 1.1 of their article (Behrisch, Cojaa-Oghlan & Kang 2014), published in issue 23 (3). The text below explains the changes required.

Given a graph G and a bijection f : E(G) → {1, 2,…,e(G)}, we say that a trail/path in G is f-increasing if the labels of consecutive edges of this trail/path form an increasing sequence. More than 40 years ago Chvátal and Komlós raised the question of providing worst-case estimates of the length of the longest increasing trail/path over all edge orderings of Kn. The case of a trail was resolved by Graham and Kleitman, who proved that the answer is n-1, and the case of a path is still wide open. Recently Lavrov and Loh proposed studying the average-case version of this problem, in which the edge ordering is chosen uniformly at random. They conjectured (and Martinsson later proved) that such an ordering with high probability (w.h.p.) contains an increasing Hamilton path.

In this paper we consider the random graph G = Gn,p with an edge ordering chosen uniformly at random. In this setting we determine w.h.p. the asymptotics of the number of edges in the longest increasing trail. In particular we prove an average-case version of the result of Graham and Kleitman, showing that the random edge ordering of Kn has w.h.p. an increasing trail of length (1-o(1))en, and that this is tight. We also obtain an asymptotically tight result for the length of the longest increasing path for random Erdős-Renyi graphs with p = o(1).

The notion of the capacity of a polynomial was introduced by Gurvits around 2005, originally to give drastically simplified proofs of the van der Waerden lower bound for permanents of doubly stochastic matrices and Schrijver’s inequality for perfect matchings of regular bipartite graphs. Since this seminal work, the notion of capacity has been utilised to bound various combinatorial quantities and to give polynomial-time algorithms to approximate such quantities (e.g. the number of bases of a matroid). These types of results are often proven by giving bounds on how much a particular differential operator can change the capacity of a given polynomial. In this paper, we unify the theory surrounding such capacity-preserving operators by giving tight capacity preservation bounds for all nondegenerate real stability preservers. We then use this theory to give a new proof of a recent result of Csikvári, which settled Friedland’s lower matching conjecture.

For fixed positive integers $k,q,r$ with $q$ a prime power and large $m$, we investigate matrices with $m$ rows and a maximum number $N_q (m,k,r)$ of columns, such that each column contains at most $r$ nonzero entries from the finite field $GF(q)$ and any $k$ columns are linearly independent over $GF(q)$. For even integers $k \geq 2$ we obtain the lower bounds $N_q(m,k,r) = \Omega (m^{kr/(2(k-1))})$, and $N_q(m,k,r) = \Omega (m^{((k-1)r)/(2(k-2))})$ for odd $k \geq 3$. For $k=2^i$ we show that $N_q(m,k,r) = \Theta ( m^{kr/(2(k-1))})$ if $\gcd(k-1,r) = k-1$, while for arbitrary even $k \geq 4$ with $\gcd(k-1,r) =1$ we have $N_q(m,k,r) = \Omega (m^{kr/(2(k-1))} \cdot (\log m)^{1/(k-1)})$. Matrices which fulfil these lower bounds can be found in polynomial time. Moreover, for $\Char (GF(q)) > 2 $ we obtain $N_q(m,4,r) = \Theta (m^{\lceil 4r/3\rceil/2})$, while for $\Char (GF(q)) = 2$ we can only show that $N_q(m,4,r) = O (m^{\lceil 4r/3\rceil/2})$. Our results extend and complement earlier results from [7, 18], where the case $q=2$ was considered.

An analytic study is made of the correlation structure of Tausworthe and linear congruential random number generators. The former case is analyzed by the bit mask correlations recently introduced by Compagner. The latter is studied first by an extension to word masks, which include spectral test coefficients as special cases, and then by the bit mask procedure. Although low order bit mask coefficients vanish in both cases, the Tausworthe generator appears to produce a substantially smaller non-vanishing correlation set for large masks – but with larger correlation values – than does the linear congruential.

The number of excluded minors for the class of graphs with path-width at most two is very large. To give a practical characterization of the obstructions, we introduce some operations which preserve path-width at most two. We give a list of ten graphs such that any graph with path-width more than two can be reduced – by taking minors and applying our operations – to one of the graphs on our list. We think that our operations and excluded substructures give a far more transparent description of the class of graphs with path-width at most two than Kinnersley and Langston's characterization by 110 excluded minors (see [4]).

A graph G arrows a graph H if in every 2-edge-colouring of G there exists a monochromatic copy of H. Schelp had the idea that if the complete graph $K_n$ arrows a small graph H, then every ‘dense’ subgraph of $K_n$ also arrows H, and he outlined some problems in this direction. Our main result is in this spirit. We prove that for every sufficiently large n, if $n = 3t+r$ where $r \in \{0,1,2\}$ and G is an n-vertex graph with $\delta(G) \ge (3n-1)/4$ , then for every 2-edge-colouring of G, either there are cycles of every length $\{3, 4, 5, \dots, 2t+r\}$ of the same colour, or there are cycles of every even length $\{4, 6, 8, \dots, 2t+2\}$ of the samecolour.

Our result is tight in the sense that no longer cycles (of length $>2t+r$ ) can be guaranteed and the minimum degree condition cannot be reduced. It also implies the conjecture of Schelp that for every sufficiently large n, every $(3t-1)$ -vertex graph G with minimum degree larger than $3|V(G)|/4$ arrows the path $P_{2n}$ with 2n vertices. Moreover, it implies for sufficiently large n the conjecture by Benevides, Łuczak, Scott, Skokan and White that for $n=3t+r$ where $r \in \{0,1,2\}$ and every n-vertex graph G with $\delta(G) \ge 3n/4$ , in each 2-edge-colouring of G there exists a monochromatic cycle of length at least $2t+r$ .

Suppose each vertex of a bipartite multigraph (with partition (X, Y)) is assigned a set of colours; we say this colour scheme is feasible if the edges of the graph can be properly coloured so that each receives a colour in both of its endpoints' sets. We prove various results showing that certain types of colour scheme are always feasible. For instance, we prove that if the colour scheme obtained by assigning the set {1,…, d(x)} of colours to each vertex x of X and the set T = {1,…, t} (t < Δ(X)) to each vertex of Y is feasible, then so is every colour scheme where each vertex x of X gets d(x) colours from T and each vertex of Y gets the set T.