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

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.

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.

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 protection number of a vertex $v$ in a tree is the length of the shortest path from $v$ to any leaf contained in the maximal subtree where $v$ is the root. In this paper, we determine the distribution of the maximum protection number of a vertex in simply generated trees, thereby refining a recent result of Devroye, Goh, and Zhao. Two different cases can be observed: if the given family of trees allows vertices of outdegree $1$, then the maximum protection number is on average logarithmic in the tree size, with a discrete double-exponential limiting distribution. If no such vertices are allowed, the maximum protection number is doubly logarithmic in the tree size and concentrated on at most two values. These results are obtained by studying the singular behaviour of the generating functions of trees with bounded protection number. While a general distributional result by Prodinger and Wagner can be used in the first case, we prove a variant of that result in the second case.

We give algorithms for approximating the partition function of the ferromagnetic $q$-color Potts model on graphs of maximum degree $d$. Our primary contribution is a fully polynomial-time approximation scheme for $d$-regular graphs with an expansion condition at low temperatures (that is, bounded away from the order-disorder threshold). The expansion condition is much weaker than in previous works; for example, the expansion exhibited by the hypercube suffices. The main improvements come from a significantly sharper analysis of standard polymer models; we use extremal graph theory and applications of Karger’s algorithm to count cuts that may be of independent interest. It is #BIS-hard to approximate the partition function at low temperatures on bounded-degree graphs, so our algorithm can be seen as evidence that hard instances of #BIS are rare. We also obtain efficient algorithms in the Gibbs uniqueness region for bounded-degree graphs. While our high-temperature proof follows more standard polymer model analysis, our result holds in the largest-known range of parameters $d$ and $q$.