In a set of even cardinality n, each member ranks all the others in order of preference. A stable matching is a partition of the set into n/2 pairs, with the property that no two unpaired members both prefer each other to their partners under matching. It is known that for some problem instances no stable matching exists. In 1985, Irving found an O(n2) two-phase algorithm that would determine, for any instance, whether a stable matching exists, and if so, would find such a matching. Recently, Tan proved that Irving's algorithm, with a modified second phase, always finds a stable cyclic partition of the members set, which is a stable matching when each cycle has length two. In this paper we study a likely behavior of the algorithm under the assumption that an instance of the ranking system is chosen uniformly at random. We prove that the likely number of basic steps, i.e. the individual proposals in the first phase and the rotation eliminations, involving subsets of members in the second phase, is O(n log n), and that the likely size of a rotation is O((n log n)1/2). We establish a ‘hyperbola law’ analogous to our past result on stable marriages. It states that at every step of the second phase, the product of the rank of proposers and the rank of proposal holders is asymptotic, in probability, to n3. We show that every stable cyclic partition is likely to be almost a stable matching, in the sense that at most O((n log n)1/2) members can be involved in the cycles of length three or more.

We generalize and unify results on parametrized and coloured Tutte polynomials of graphs and matroids due to Zaslavsky, and Bollobás and Riordan. We give a generalized Zaslavsky–Bollobás–Riordan theorem that characterizes parametrized contraction–deletion functions on minor-closed classes of matroids, as well as the modifications necessary to apply the discussion to classes of graphs. In general, these parametrized Tutte polynomials do not satisfy analogues of all the familiar properties of the classical Tutte polynomial. We give conditions under which they do satisfy corank-nullity formulas, and also conditions under which they reflect the structure of series-parallel connections.

Let kp(G) denote the number of complete subgraphs of order p in the graph G. Bollobás proved that any real linear combination of the form [sum ]apkp(G) attains its maximum on a complete multipartite graph. We show that the same is true for a linear combination of the form [sum ]apkp(G) +bpkp(G¯), provided bp[ges ]0 for every p.

We show that if a graph contains few copies of a given graph, then its edges are distributed rather unevenly.

In particular, for all $\varepsilon > 0$ and $r\geq2$, there exist $\xi =\xi (\varepsilon,r) > 0$ and $k=k (\varepsilon,r)$ such that, if $n$ is sufficiently large and $G=G(n)$ is a graph with fewer than $\xi n^{r}$$r$-cliques, then there exists a partition $V(G) =\cup_{i=0}^{k}V_{i}$ such that \[ \vert V_{i}\vert =\lfloor n/k\rfloor \quad \text{and} \quad e(W_{i}) <\varepsilon\vert V_{i}\vert ^{2}\] for every $i\in [k]$.

We deduce the following slightly stronger form of a conjecture of Erdős.

For all $c>0$ and $r\geq3$, there exist $\xi=\xi (c,r) >0$ and $\beta=\beta(c,r)>0$ such that, if $n$ is sufficiently large and $G=G(n,\lceil cn^{2} \rceil)$ is a graph with fewer than $\xi n^{r}$$r$-cliques, then there exists a partition $V(G) =V_{1}\cup V_{2}$ with $ \vert V_{1} \vert = \lfloor n/2 \rfloor $ and $\vert V_{2} \vert = \lceil n/2 \rceil $ such that \[ e(V_{1},V_{2}) > (1/2+\beta) e (G).\]

Let k ⩾ 3 be a fixed integer and let Zk(G) be the number of k-colourings of the graph G. For certain values of the average degree, the random variable Zk(G(n, m)) is known to be concentrated in the sense that $\tfrac{1}{n}(\ln Z_k(G(n,m))-\ln\Erw[Z_k(G(n,m))])$ converges to 0 in probability (Achlioptas and Coja-Oghlan, Proc. FOCS 2008). In the present paper we prove a significantly stronger concentration result. Namely, we show that for a wide range of average degrees, $\tfrac{1}{\omega}(\ln Z_k(G(n,m))-\ln\Erw[Z_k(G(n,m))])$ converges to 0 in probability for any diverging function $\omega=\omega(n)\ra\infty$. For k exceeding a certain constant k0 this result covers all average degrees up to the so-called condensation phase transitiondk,con, and this is best possible. As an application, we show that the experiment of choosing a k-colouring of the random graph G(n,m) uniformly at random is contiguous with respect to the so-called ‘planted model’.

Let (Xn) be a residual allocation model with i.i.d. residual fractions Un: For W a random variable with values in [0; 1] and independent of (Xn), we define another sequence (Yn) by setting

(formula here)

Under minor regularity assumptions we show that (Xn) and (Yn) have the same probability law if and only if this law is a GEM distribution. In this case, the distribution of W and the Uns is Beta(1; θ) for some θ > 0.

We show that the random insertion method for the traveling salesman problem (TSP) may produce a tour Ω(log log n/log log log n) times longer than the optimal tour. The lower bound holds even in the Euclidean Plane. This is in contrast to the fact that the random insertion method performs extremely well in practice. In passing, we show that other insertion methods may produce tours Ω(log n/log log n) times longer than the optimal one. No non-constant lower bounds were previously known.

Notions of deletion and contraction for the class of set functions from finite sets into the integers are defined. An operation on a subclass of such set functions is a function from the subclass into itself that preserves ground sets and respects isomorphism. The operations on set functions that interchange deletion and contraction are characterised, as are those with the further property of being involutary. Similar results are given for polymatroids. There is a unique involutary operation on the class of k-polymatroids that interchanges deletion and contraction. The results generalise those of Kung [3].

We prove that there is a constant c such that, for each positive integer k, every (2k + 1) × (2k + 1) array A on the symbols (1,. . .,2k+1) with at most c(2k+1) symbols in every cell, and each symbol repeated at most c(2k+1) times in every row and column is avoidable; that is, there is a (2k+1) × (2k+1) Latin square S on the symbols 1,. . .,2k+1 such that, for each i,j ∈ {1,. . .,2k+1}, the symbol in position (i,j) of S does not appear in the corresponding cell in A. This settles the last open case of a conjecture by Häggkvist. Using this result, we also show that there is a constant ρ, such that, for any positive integer n, if each cell in an n × n array B is assigned a set of m ≤ ρ n symbols, where each set is chosen independently and uniformly at random from {1,. . .,n}, then the probability that B is avoidable tends to 1 as n → ∞.

Let G1 and G2 be graphs of order n with maximum degree Δ1 and Δ2, respectively. G1 and G2 are said to pack if there exist injective mappings of the vertex sets into [n], such that the images of the edge sets do not intersect. Sauer and Spencer showed that if , then G1 and G2 pack. We extend this result by showing that if , then G1 and G2 do not pack if and only if one of G1 or G2 is a perfect matching and the other either is with odd or contains .

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.

We consider the following random graph process: starting with n isolated vertices, add edges uniformly at random provided no such edge creates a copy of C4. We show that, with probability tending to 1 as n → ∞, the final graph produced by this process has maximum degree O((nlogn)1/3) and consequently size O(n4/3(logn)1/3), which are sharp up to constants. This confirms conjectures of Bohman and Keevash and of Osthus and Taraz, and improves upon previous bounds due to Bollobás and Riordan and Osthus and Taraz.

It is shown that unless NP collapses to random polynomial time RP, there can be no fully polynomial randomised approximation scheme for the antiferromagnetic version of the Q-state Potts model.

We study the minimum degree necessary to guarantee the existence of perfect and almost-perfect triangle-tilings in an n-vertex graph G with sublinear independence number. In this setting, we show that if δ(G) ≥ n/3 + o(n), then G has a triangle-tiling covering all but at most four vertices. Also, for every r ≥ 5, we asymptotically determine the minimum degree threshold for a perfect triangle-tiling under the additional assumptions that G is Kr-free and n is divisible by 3.

A set A of positive integers is a Bh-set if all the sums of the form a1 + . . . + ah, with a1,. . .,ah ∈ A and a1 ⩽ . . . ⩽ ah, are distinct. We provide asymptotic bounds for the number of Bh-sets of a given cardinality contained in the interval [n] = {1,. . .,n}. As a consequence of our results, we address a problem of Cameron and Erdős (1990) in the context of Bh-sets. We also use these results to estimate the maximum size of a Bh-sets contained in a typical (random) subset of [n] with a given cardinality.

Let k be a positive integer and G a finite abelian group of order n, where n[ges ]k2−4k+8. Then every sequence of 2n−¼k2+k−2 elements in G assuming k distinct values has an n-subsequence with sum zero. This settles a conjecture of Bialostocki and Lotspeich.

A property of finite graphs is called non-deterministically testable if it has a ‘certificate’ such that once the certificate is specified, its correctness can be verified by random local testing. In this paper we study certificates that consist of one or more unary and/or binary relations on the nodes, in the case of dense graphs. Using the theory of graph limits, we prove that non-deterministically testable properties are also deterministically testable.

The vertex-nullity interlace polynomial of a graph, described by Arratia, Bollobás and Sorkin in [3] as evolving from questions of DNA sequencing, and extended to a two-variable interlace polynomial by the same authors in [5], evokes many open questions. These include relations between the interlace polynomial and the Tutte polynomial and the computational complexity of the vertex-nullity interlace polynomial. Here, using the medial graph of a planar graph, we relate the one-variable vertex-nullity interlace polynomial to the classical Tutte polynomial when x=y, and conclude that, like the Tutte polynomial, it is in general #P-hard to compute. We also show a relation between the two-variable interlace polynomial and the topological Tutte polynomial of Bollobás and Riordan in [13].

We define the γ invariant as the coefficient of x1 in the vertex-nullity interlace polynomial, analogously to the β invariant, which is the coefficientof x1 in the Tutte polynomial. We then turn to distance hereditary graphs, characterized by Bandelt and Mulder in [9] as being constructed by a sequence ofadding pendant and twin vertices, and show that graphs in this class have γ invariant of 2n+1 when n true twins are added intheir construction. We furthermore show that bipartite distance hereditary graphs are exactly the class of graphs with γ invariant 2, just as the series-parallel graphs are exactly the class of graphs with β invariant 1. In addition, we show that a bipartite distance hereditary graph arises precisely as the circle graph of an Euler circuitin the oriented medial graph of a series-parallel graph. From this we conclude that the vertex-nullity interlace polynomial is polynomial time to compute for bipartite distancehereditary graphs, just as the Tutte polynomial is polynomial time to compute for series-parallel graphs.

We consider the minimum-weight path between any pair of nodes of the n-vertex complete graph in which the weights of the edges are i.i.d. exponentially distributed random variables. We show that the longest of these minimum-weight paths has about α* log n edges, where α* ≈ 3.5911 is the unique solution of the equation α log α − α = 1. This answers a question posed by Janson [8].

Let $s$ and $t$ be integers satisfying $s \geq 2$ and $t \geq 2$. Let $S$ be a tree of size $s$, and let $P_t$ be the path of length $t$. We show in this paper that, for every edge-colouring of the complete graph on $n$ vertices, where $n=224(s-1)^2t$, there is either a monochromatic copy of $S$ or a rainbow copy of $P_t$. So, in particular, the number of vertices needed grows only linearly in $t$.