Starting from an n-by-n matrix of zeros, choose uniformly random zero entries and change them to ones, one at a time, until the matrix becomes invertible. We show that with probability tending to one as n → ∞, this occurs at the very moment the last zero row or zero column disappears. We prove a related result for random symmetric Bernoulli matrices, and give quantitative bounds for some related problems. These results extend earlier work by Costello and Vu [10].

We study the diameter of a family of random graphs on the torus that can be used to model wireless networks. In the random connection model two points x and y are connected with probability g(y−x), where g is a given function. We prove that the diameter of the graph is bounded by a constant, which depends only on ‖g‖1, with high probability as the number of vertices in the graph tends to infinity.

Let H be a graph, and let CH(G) be the number of (subgraph isomorphic) copies of H contained in a graph G. We investigate the fundamental problem of estimating CH(G). Previous results cover only a few specific instances of this general problem, for example the case when H has degree at most one (the monomer-dimer problem). In this paper we present the first general subcase of the subgraph isomorphism counting problem, which is almost always efficiently approximable. The results rely on a new graph decomposition technique. Informally, the decomposition is a labelling of the vertices such that every edge is between vertices with different labels, and for every vertex all neighbours with a higher label have identical labels. The labelling implicitly generates a sequence of bipartite graphs, which permits us to break the problem of counting embeddings of large subgraphs into that of counting embeddings of small subgraphs. Using this method, we present a simple randomized algorithm for the counting problem. For all decomposable graphs H and all graphs G, the algorithm is an unbiased estimator. Furthermore, for all graphs H having a decomposition where each of the bipartite graphs generated is small and almost all graphs G, the algorithm is a fully polynomial randomized approximation scheme.

We show that the graph classes of H for which we obtain a fully polynomial randomized approximation scheme for almost all G includes graphs of degree at most two, bounded-degree forests, bounded-width grid graphs, subdivision of bounded-degree graphs, and major subclasses of outerplanar graphs, series-parallel graphs and planar graphs of large girth, whereas unbounded-width grid graphs are excluded. Moreover, our general technique can easily be applied to proving many more similar results.

We prove that the number of 1324-avoiding permutations of length n is less than $(7+4\sqrt{3})^n$. The novelty of our method is that we injectively encode such permutations by a pair of words of length n over a finite alphabet that avoid a given factor.

We consider a recently defined notion of k-abelian equivalence of words byconcentrating on avoidance problems. The equivalence class of a word depends on thenumbers of occurrences of different factors of length k for a fixed naturalnumber k andthe prefix of the word. We have shown earlier that over a ternary alphabet k-abelian squares cannot beavoided in pure morphic words for any natural number k. Nevertheless,computational experiments support the conjecture that even 3-abelian squares can beavoided over ternary alphabets. In this paper we establish the first avoidance resultshowing that by choosing k to be large enough we have an infinitek-abeliansquare-free word over three letter alphabet. In addition, this word can be obtained as amorphic image of a pure morphic word.

We consider numeration systems with base β and −β, for quadratic Pisot numbers β and focus on comparingthe combinatorial structure of the sets Zβ and Z− β of numberswith integer expansion in base β, resp. − β. Our main result is the comparison of languagesof infinite words uβ andu−β coding the ordering of distances betweenconsecutive β- and (−β)-integers. It turns out that for a class of rootsβ ofx2 −mx − m, the languages coincide,while for other quadratic Pisot numbers the language of uβ can be identified onlywith the language of a morphic image of u− β. We also study thegroup structure of (−β)-integers.

The spread of a connected graph G was introduced by Alon, Boppana and Spencer [1], and measures how tightly connected the graph is. It is defined as the maximum over all Lipschitz functions f on V(G) of the variance of f(X) when X is uniformly distributed on V(G). We investigate the spread for certain models of sparse random graph, in particular for random regular graphs G(n,d), for Erdős–Rényi random graphs Gn,p in the supercritical range p>1/n, and for a ‘small world’ model. For supercritical Gn,p, we show that if p=c/n with c>1 fixed, then with high probability the spread of the giant component is bounded, and we prove corresponding statements for other models of random graphs, including a model with random edge lengths. We also give lower bounds on the spread for the barely supercritical case when p=(1+o(1))/n. Further, we show that for d large, with high probability the spread of G(n,d) becomes arbitrarily close to that of the complete graph $\mathsf{K}_n$.

We extend the conductance and canonical paths methods to the setting of general finite Markov chains, including non-reversible non-lazy walks. The new path method is used to show that a known bound for the mixing time of a lazy walk on a Cayley graph with a symmetric generating set also applies to the non-lazy non-symmetric case, often even when there is no holding probability.

Pattern avoidance is an important topic in combinatorics on words which dates back to thebeginning of the twentieth century when Thue constructed an infinite word over a ternaryalphabet that avoids squares, i.e., a word with no two adjacent identicalfactors. This result finds applications in various algebraic contexts where more generalpatterns than squares are considered. On the other hand, Erdős raised the question as towhether there exists an infinite word that avoids abelian squares, i.e.,a word with no two adjacent factors being permutations of one another. Although thisquestion was answered affirmately years later, knowledge of abelian pattern avoidance israther limited. Recently, (abelian) pattern avoidance was initiated in the more generalframework of partial words, which allow for undefined positions called holes. In thispaper, we show that any pattern p with n> 3 distinct variables oflength at least 2n is abelian avoidable by a partial wordwith infinitely many holes, the bound on the length of p being tight. We completethe classification of all the binary and ternary patterns with respect to non-trivialabelian avoidability, in which no variable can be substituted by only one hole. We alsoinvestigate the abelian avoidability indices of the binary and ternary patterns.

In the quarter plane, five lattice path models with unit steps have resisted the otherwise general approach featured in recent works by Fayolle, Kurkova and Raschel. Here we consider these five models, called the singular models, and prove that the univariate generating functions marking the number of walks of a given length are not D-finite. Furthermore, we provide exact and asymptotic enumerative formulas for the number of such walks, and describe an efficient algorithm for exact enumeration.