A collection of graphs is nearly disjoint if every pair of them intersects in at most one vertex. We prove that if $G_1, \dots, G_m$ are nearly disjoint graphs of maximum degree at most $D$, then the following holds. For every fixed $C$, if each vertex $v \in \bigcup _{i=1}^m V(G_i)$ is contained in at most $C$ of the graphs $G_1, \dots, G_m$, then the (list) chromatic number of $\bigcup _{i=1}^m G_i$ is at most $D + o(D)$. This result confirms a special case of a conjecture of Vu and generalizes Kahn’s bound on the list chromatic index of linear uniform hypergraphs of bounded maximum degree. In fact, this result holds for the correspondence (or DP) chromatic number and thus implies a recent result of Molloy and Postle, and we derive this result from a more general list colouring result in the setting of ‘colour degrees’ that also implies a result of Reed and Sudakov.

In 2003, Bohman, Frieze, and Martin initiated the study of randomly perturbed graphs and digraphs. For digraphs, they showed that for every $\alpha \gt 0$, there exists a constant $C$ such that for every $n$-vertex digraph of minimum semi-degree at least $\alpha n$, if one adds $Cn$ random edges then asymptotically almost surely the resulting digraph contains a consistently oriented Hamilton cycle. We generalize their result, showing that the hypothesis of this theorem actually asymptotically almost surely ensures the existence of every orientation of a cycle of every possible length, simultaneously. Moreover, we prove that we can relax the minimum semi-degree condition to a minimum total degree condition when considering orientations of a cycle that do not contain a large number of vertices of indegree $1$. Our proofs make use of a variant of an absorbing method of Montgomery.

Since the 1960s Mastermind has been studied for the combinatorial and information-theoretical interest the game has to offer. Many results have been discovered starting with Erdős and Rényi determining the optimal number of queries needed for two colours. For $k$ colours and $n$ positions, Chvátal found asymptotically optimal bounds when $k \le n^{1-\varepsilon }$. Following a sequence of gradual improvements for $k\geq n$ colours, the central open question is to resolve the gap between $\Omega (n)$ and $\mathcal{O}(n\log \log n)$ for $k=n$. In this paper, we resolve this gap by presenting the first algorithm for solving $k=n$ Mastermind with a linear number of queries. As a consequence, we are able to determine the query complexity of Mastermind for any parameters $k$ and $n$.

Given a graph $H$, let us denote by $f_\chi (H)$ and $f_\ell (H)$, respectively, the maximum chromatic number and the maximum list chromatic number of $H$-minor-free graphs. Hadwiger’s famous colouring conjecture from 1943 states that $f_\chi (K_t)=t-1$ for every $t \ge 2$. A closely related problem that has received significant attention in the past concerns $f_\ell (K_t)$, for which it is known that $2t-o(t) \le f_\ell (K_t) \le O(t (\!\log \log t)^6)$. Thus, $f_\ell (K_t)$ is bounded away from the conjectured value $t-1$ for $f_\chi (K_t)$ by at least a constant factor. The so-called $H$-Hadwiger’s conjecture, proposed by Seymour, asks to prove that $f_\chi (H)={\textrm{v}}(H)-1$ for a given graph $H$ (which would be implied by Hadwiger’s conjecture).

In this paper, we prove several new lower bounds on $f_\ell (H)$, thus exploring the limits of a list colouring extension of $H$-Hadwiger’s conjecture. Our main results are:

• For every $\varepsilon \gt 0$ and all sufficiently large graphs $H$ we have $f_\ell (H)\ge (1-\varepsilon )({\textrm{v}}(H)+\kappa (H))$, where $\kappa (H)$ denotes the vertex-connectivity of $H$.

• For every $\varepsilon \gt 0$ there exists $C=C(\varepsilon )\gt 0$ such that asymptotically almost every $n$-vertex graph $H$ with $\left \lceil C n\log n\right \rceil$ edges satisfies $f_\ell (H)\ge (2-\varepsilon )n$.

The first result generalizes recent results on complete and complete bipartite graphs and shows that the list chromatic number of $H$-minor-free graphs is separated from the desired value of $({\textrm{v}}(H)-1)$ by a constant factor for all large graphs $H$ of linear connectivity. The second result tells us that for almost all graphs $H$ with superlogarithmic average degree $f_\ell (H)$ is separated from $({\textrm{v}}(H)-1)$ by a constant factor arbitrarily close to $2$. Conceptually these results indicate that the graphs $H$ for which $f_\ell (H)$ is close to the conjectured value $({\textrm{v}}(H)-1)$ for $f_\chi (H)$ are typically rather sparse.

We show that for every $n\in \mathbb N$ and $\log n\le d\lt n$, if a graph $G$ has $N=\Theta (dn)$ vertices and minimum degree $(1+o(1))\frac{N}{2}$, then it contains a spanning subdivision of every $n$-vertex $d$-regular graph.

A set of vertices in a graph is a Hamiltonian subset if it induces a subgraph containing a Hamiltonian cycle. Kim, Liu, Sharifzadeh, and Staden proved that for large $d$, among all graphs with minimum degree $d$, $K_{d+1}$ minimises the number of Hamiltonian subsets. We prove a near optimal lower bound that takes also the order and the structure of a graph into account. For many natural graph classes, it provides a much better bound than the extremal one ($\approx 2^{d+1}$). Among others, our bound implies that an $n$-vertex $C_4$-free graph with minimum degree $d$ contains at least $n2^{d^{2-o(1)}}$ Hamiltonian subsets.

Let $\mathcal{F}$ be an intersecting family. A $(k-1)$-set $E$ is called a unique shadow if it is contained in exactly one member of $\mathcal{F}$. Let ${\mathcal{A}}=\{A\in \binom{[n]}{k}\colon |A\cap \{1,2,3\}|\geq 2\}$. In the present paper, we show that for $n\geq 28k$, $\mathcal{A}$ is the unique family attaining the maximum size among all intersecting families without unique shadow. Several other results of a similar flavour are established as well.