We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

We survey recent advances in the theory of graph and hypergraph decompositions, with a focus on extremal results involving minimum degree conditions. We also collect a number of intriguing open problems, and formulate new ones.

Extremal aspects of graph and hypergraph decomposition problems

On Hamiltonian cycles in hypergraphs with dense link graphs

We employ the absorbing-path method in order to prove two results regarding the emergence of tight Hamilton cycles in the so-called two-path or cherry-quasirandom 3-graphs.
 Our first result asserts that for any fixed real α > 0, cherry-quasirandom 3-graphs of sufficiently large order n having minimum 2-degree at least α(n – 2) have a tight Hamilton cycle.
 Our second result concerns the minimum 1-degree sufficient for such 3-graphs to have a tight Hamilton cycle. Roughly speaking, we prove that for every d, α > 0 satisfying d + α > 1, any sufficiently large n-vertex such 3-graph H of density d and minimum 1-degree at least has a tight Hamilton cycle.

/pdf/tight-hamilton-cycles-in-cherry-quasirandom-3-uniform-1d7p6v2smq.pdf

Tight Hamilton cycles in cherry-quasirandom 3-uniform hypergraphs

We show that every $k$-uniform hypergraph on $n$ vertices whose minimum $(k-2)$-degree is at least $(5/9+o(1))n^2/2$ contains a Hamiltonian cycle. A construction due to Han and Zhao shows that this minimum degree condition is optimal. The same result was proved independently by Lang and Sahueza-Matamala.

pdf/on-hamiltonian-cycles-in-hypergraphs-with-dense-link-graphs-15ermnw5p3.pdf

We study sufficient conditions for the existence of Hamilton cycles in uniformly dense $3$-uniform hypergraphs. Problems of this type were first considered by Lenz, Mubayi, and Mycroft for loose Hamilton cycles and Aigner-Horev and Levy considered it for tight Hamilton cycles for a fairly strong notion of uniformly dense hypergraphs. We focus on tight cycles and obtain optimal results for a weaker notion of uniformly dense hypergraphs. 
We show that if an $n$-vertex $3$-uniform hypergraph $H=(V,E)$ has the property that for any set of vertices $X$ and for any collection $P$ of pairs of vertices, the number of hyperedges composed by a pair belonging to $P$ and one vertex from $X$ is at least $(1/4+o(1))|X||P| - o(|V|^3)$ and $H$ has minimum vertex degree at least $\Omega(|V|^2)$, then $H$ contains a tight Hamilton cycle. A probabilistic construction shows that the constant $1/4$ is optimal in this context.

Localised codegree conditions for tight Hamilton cycles in 3-uniform hypergraphs.

We study sufficient conditions for the existence of Hamilton cycles in uniformly dense $3$-uniform hypergraphs. Problems of this type were first considered by Lenz, Mubayi, and Mycroft for loose Hamilton cycles and Aigner-Horev and Levy considered it for tight Hamilton cycles for a fairly strong notion of uniformly dense hypergraphs. We focus on tight cycles and obtain optimal results for a weaker notion of uniformly dense hypergraphs. 
We show that if an $n$-vertex $3$-uniform hypergraph $H=(V,E)$ has the property that for any set of vertices $X$ and for any collection $P$ of pairs of vertices, the number of hyperedges composed by a pair belonging to $P$ and one vertex from $X$ is at least $(1/4+o(1))|X||P| - o(|V|^3)$ and $H$ has minimum vertex degree at least $\Omega(|V|^2)$, then $H$ contains a tight Hamilton cycle. A probabilistic construction shows that the constant $1/4$ is optimal in this context.

/pdf/localised-codegree-conditions-for-tight-hamiltonian-cycles-5cio6n0tfu.pdf

Localised codegree conditions for tight Hamiltonian cycles in 3-uniform hypergraphs

We show that $3$-graphs on $n$ vertices whose codegree is at least $(2/3 + o(1))n$ can be decomposed into tight cycles and admit Euler tours, subject to the trivial necessary divisibility conditions. We also provide a construction showing that our bounds are best possible up to the $o(1)$ term. All together, our results answer in the negative some recent questions of Glock, Joos, Kuhn, and Osthus.

Cycle decompositions in $3$-uniform hypergraphs

For a fixed poset $P$, a family $\mathcal F$ of subsets of $[n]$ is induced $P$-saturated if $\mathcal F$ does not contain an induced copy of $P$, but for every subset $S$ of $[n]$ such that $ S\not \in \mathcal F$, then $P$ is an induced subposet of $\mathcal F \cup \{S\}$. The size of the smallest such family $\mathcal F$ is denoted by $\text{sat}^* (n,P)$. Keszegh, Lemons, Martin, P\‘alv\"olgyi and Patk\‘os [Journal of Combinatorial Theory Series A, 2021] proved that there is a dichotomy of behaviour for this parameter: given any poset $P$, either $\text{sat}^* (n,P)=O(1)$ or $\text{sat}^* (n,P)\geq \log _2 n$. We improve this general result showing that either $\text{sat}^* (n,P)=O(1)$ or $\text{sat}^* (n,P) \geq 2 \sqrt{n-2}$. Our proof makes use of a Tur\‘an-type result for digraphs. Curiously, it remains open as to whether our result is essentially best possible or not. On the one hand, a conjecture of Ivan states that for the so-called diamond poset $\Diamond$ we have $\text{sat}^* (n,\Diamond)=\Theta (\sqrt{n})$; so if true this conjecture implies our result is tight up to a multiplicative constant. On the other hand, a conjecture of Keszegh, Lemons, Martin, P\‘alv\"olgyi and Patk\‘os states that given any poset $P$, either $\text{sat}^* (n,P)=O(1)$ or $\text{sat}^* (n,P)\geq n+1$. We prove that this latter conjecture is true for a certain class of posets $P$.

https://journals.muni.cz/eurocomb/article/download/35597/30333

A general bound for the induced poset saturation problem

In 2003, Bohman, Frieze, and Martin initiated the study of randomly perturbed graphs and digraphs. For digraphs, they showed that for every $\alpha>0$, 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 $1$.

https://journals.muni.cz/eurocomb/article/download/35543/30394

Cycles of every length and orientation in randomly perturbed digraphs

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