The bipartite independence number of a graph $G$, denoted as $\tilde \alpha (G)$, is the minimal number $k$ such that there exist positive integers $a$ and $b$ with $a+b=k+1$ with the property that for any two disjoint sets $A,B\subseteq V(G)$ with $|A|=a$ and $|B|=b$, there is an edge between $A$ and $B$. McDiarmid and Yolov showed that if $\delta (G)\geq \tilde \alpha (G)$ then $G$ is Hamiltonian, extending the famous theorem of Dirac which states that if $\delta (G)\geq |G|/2$ then $G$ is Hamiltonian. In 1973, Bondy showed that, unless $G$ is a complete bipartite graph, Dirac’s Hamiltonicity condition also implies pancyclicity, i.e., existence of cycles of all the lengths from $3$ up to $n$. In this paper, we show that $\delta (G)\geq \tilde \alpha (G)$ implies that $G$ is pancyclic or that $G=K_{\frac{n}{2},\frac{n}{2}}$, thus extending the result of McDiarmid and Yolov, and generalizing the classic theorem of Bondy.

The $d$-process generates a graph at random by starting with an empty graph with $n$ vertices, then adding edges one at a time uniformly at random among all pairs of vertices which have degrees at most $d-1$ and are not mutually joined. We show that, in the evolution of a random graph with $n$ vertices under the $d$-process with $d$ fixed, with high probability, for each $j \in \{0,1,\dots,d-2\}$, the minimum degree jumps from $j$ to $j+1$ when the number of steps left is on the order of $\ln (n)^{d-j-1}$. This answers a question of Ruciński and Wormald. More specifically, we show that, when the last vertex of degree $j$ disappears, the number of steps left divided by $\ln (n)^{d-j-1}$ converges in distribution to the exponential random variable of mean $\frac{j!}{2(d-1)!}$; furthermore, these $d-1$ distributions are independent.

Given a graph $F$, we consider the problem of determining the densest possible pseudorandom graph that contains no copy of $F$. We provide an embedding procedure that improves a general result of Conlon, Fox, and Zhao which gives an upper bound on the density. In particular, our result implies that optimally pseudorandom graphs with density greater than $n^{-1/3}$ must contain a copy of the Peterson graph, while the previous best result gives the bound $n^{-1/4}$. Moreover, we conjecture that the exponent $1/3$ in our bound is tight. We also construct the densest known pseudorandom $K_{2,3}$-free graphs that are also triangle-free. Finally, we give a different proof for the densest known construction of clique-free pseudorandom graphs due to Bishnoi, Ihringer, and Pepe that they have no large clique.

We investigate here the behaviour of a large typical meandric system, proving a central limit theorem for the number of components of a given shape. Our main tool is a theorem of Gao and Wormald that allows us to deduce a central limit theorem from the asymptotics of large moments of our quantities of interest.

We find asymptotics of the maximum size of a chordal subgraph in a binomial random graph $G(n,p)$, for $p=\mathrm{const}$ and $p=n^{-\alpha +o(1)}$.

We investigate the existence of a rainbow Hamilton cycle in a uniformly edge-coloured randomly perturbed digraph. We show that for every $\delta \in (0,1)$ there exists $C = C(\delta ) \gt 0$ such that the following holds. Let $D_0$ be an $n$-vertex digraph with minimum semidegree at least $\delta n$ and suppose that each edge of the union of $D_0$ with a copy of the random digraph $\mathbf{D}(n,C/n)$ on the same vertex set gets a colour in $[n]$ independently and uniformly at random. Then, with high probability, $D_0 \cup \mathbf{D}(n,C/n)$ has a rainbow directed Hamilton cycle.

This improves a result of Aigner-Horev and Hefetz ((2021) SIAM J. Discrete Math. 35(3) 1569–1577), who proved the same in the undirected setting when the edges are coloured uniformly in a set of $(1 + \varepsilon )n$ colours.

We derive a sufficient condition for a sparse random matrix with given numbers of non-zero entries in the rows and columns having full row rank. The result covers both matrices over finite fields with independent non-zero entries and $\{0,1\}$-matrices over the rationals. The sufficient condition is generally necessary as well.