Counting Hamilton cycles in sparse random directed graphs

Let D(n,p) be the random directed graph on n vertices where each of the n(n-1) possible arcs is present independently with probability p. A celebrated result of Frieze shows that if $p\ge(\log n+\omega(1))/n$ then D(n,p) typically has a directed Hamilton cycle, and this is best possible. In this paper, we obtain a strengthening of this result, showing that under the same condition, the number of directed Hamilton cycles in D(n,p) is typically $n!(p(1+o(1)))^{n}$. We also prove a hitting-time version of this statement, showing that in the random directed graph process, as soon as every vertex has in-/out-degrees at least 1, there are typically $n!(\log n/n(1+o(1)))^{n}$ directed Hamilton cycles

Hamilton decompositions of regular expanders: applications

In a recent paper, we showed that every sufficiently large regular digraph G on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge-disjoint Hamilton cycles. The main consequence of this theorem is that every regular tournament on n vertices can be decomposed into (n-1)/2 edge-disjoint Hamilton cycles, whenever n is sufficiently large. This verified a conjecture of Kelly from 1968. In this paper, we derive a number of further consequences of our result on robust outexpanders, the main ones are the following: (i) an undirected analogue of our result on robust outexpanders; (ii) best possible bounds on the size of an optimal packing of edge-disjoint Hamilton cycles in a graph of minimum degree d for a large range of values for d. (iii) a similar result for digraphs of given minimum semidegree; (iv) an approximate version of a conjecture of Nash-Williams on Hamilton decompositions of dense regular graphs; (v) the observation that dense quasi-random graphs are robust outexpanders; (vi) a verification of the `very dense' case of a conjecture of Frieze and Krivelevich on packing edge-disjoint Hamilton cycles in random graphs; (vii) a proof of a conjecture of Erdos on the size of an optimal packing of edge-disjoint Hamilton cycles in a random tournament.Comment: final version, to appear in J. Combinatorial Theory

Optimal Hamilton covers and linear arboricity for random graphs

In his seminal 1976 paper, P\'osa showed that for all $p\geq C\log n/n$, the binomial random graph $G(n,p)$ is with high probability Hamiltonian. This leads to the following natural questions, which have been extensively studied: How well is it typically possible to cover all edges of $G(n,p)$ with Hamilton cycles? How many cycles are necessary? In this paper we show that for $p\geq C\log n/n$, we can cover $G\sim G(n,p)$ with precisely $\lceil\Delta(G)/2\rceil$ Hamilton cycles. Our result is clearly best possible both in terms of the number of required cycles, and the asymptotics of the edge probability $p$, since it starts working at the weak threshold needed for Hamiltonicity. This resolves a problem of Glebov, Krivelevich and Szab\'o, and improves upon previous work of Hefetz, K\"uhn, Lapinskas and Osthus, and of Ferber, Kronenberg and Long, essentially closing a long line of research on Hamiltonian packing and covering problems in random graphs.Comment: 13 page

Robust Hamiltonicity in families of Dirac graphs

A graph is called Dirac if its minimum degree is at least half of the number of vertices in it. Joos and Kim showed that every collection $\mathbb{G}=\{G_1,\ldots,G_n\}$ of Dirac graphs on the same vertex set $V$ of size $n$ contains a Hamilton cycle transversal, i.e., a Hamilton cycle $H$ on $V$ with a bijection $\phi:E(H)\rightarrow [n]$ such that $e\in G_{\phi(e)}$ for every $e\in E(H)$. In this paper, we determine up to a multiplicative constant, the threshold for the existence of a Hamilton cycle transversal in a collection of random subgraphs of Dirac graphs in various settings. Our proofs rely on constructing a spread measure on the set of Hamilton cycle transversals of a family of Dirac graphs. As a corollary, we obtain that every collection of $n$ Dirac graphs on $n$ vertices contains at least $(cn)^{2n}$ different Hamilton cycle transversals $(H,\phi)$ for some absolute constant $c>0$. This is optimal up to the constant $c$. Finally, we show that if $n$ is sufficiently large, then every such collection spans $n/2$ pairwise edge-disjoint Hamilton cycle transversals, and this is best possible. These statements generalize classical counting results of Hamilton cycles in a single Dirac graph

Optimal path and cycle decompositions of dense quasirandom graphs

Motivated by longstanding conjectures regarding decompositions of graphs into paths and cycles, we prove the following optimal decomposition results for random graphs. Let $0<p<1$ be constant and let $G\sim G_{n,p}$. Let $odd(G)$ be the number of odd degree vertices in $G$. Then a.a.s. the following hold: (i) $G$ can be decomposed into $\lfloor\Delta(G)/2\rfloor$ cycles and a matching of size $odd(G)/2$. (ii) $G$ can be decomposed into $\max\{odd(G)/2,\lceil\Delta(G)/2\rceil\}$ paths. (iii) $G$ can be decomposed into $\lceil\Delta(G)/2\rceil$ linear forests. Each of these bounds is best possible. We actually derive (i)--(iii) from `quasirandom' versions of our results. In that context, we also determine the edge chromatic number of a given dense quasirandom graph of even order. For all these results, our main tool is a result on Hamilton decompositions of robust expanders by K\"uhn and Osthus.Comment: Some typos from the first version have been correcte

Topics in Extremal and Probabilistic Combinatorics

In this thesis, we consider a collection of problems in extremal and probabilistic combinatorics, specifically graph theory. First, we consider a close relative of the Cops and Robbers game called Revolutionaries and Spies, a two-player pursuit/evasion game devised by Beck to model network security. We show that on a ‘typical’ graph, if the second player has fewer pieces than are required to execute a particular trivial winning strategy, then the game is a first player win. Second, we consider the emergence of the square of a Hamilton cycle in a random geometric graph process, and show that typically, the exact instant at which a simple local obstacle is eliminated at every vertex, is the exact instant at which the graph becomes square Hamiltonian. This is in stark contrast to the ‘normal’ Erdos-Rényi random graph process, in which square Hamiltonicity is both not ‘local' in this sense, and occurs only once the graph is reasonably dense. Finally, we study an extremal problem concerning tournaments, that of maximising the number of oriented cycles of a fixed length. A ‘folklore’ result states that for 3-cycles one cannot do significantly better than a random tournament. More recent work shows that same is true for 5-cycles, and perhaps surprisingly that this is not true for 4-cycles. We conjecture that one can significantly beat the random tournament in expectation if and only if the length of the cycle is divisible by four, proving the ‘if’ statement, as well as a variety of new cases of the ‘only if' statement, including the case that the graph is sufficiently close to being regular.Engineering and Physical Sciences Research Council grant EP/M506394/

Packing and embedding large subgraphs

This thesis contains several embedding results for graphs in both random and non random settings. Most notably, we resolve a long standing conjecture that the threshold probability for Hamiltonicity in the random binomial subgraph of the hypercube equals $1/2$. %posed e.g.~by Bollob\'as, In Chapter 2 we obtain the following perturbation result regarding the hypercube \cQ^n: if H\subseteq\cQ^n satisfies $\delta(H)\geq\alpha n$ with $\alpha>0$ fixed and we consider a random binomial subgraph \cQ^n_p of \cQ^n with $p\in(0,1]$ fixed, then with high probability H\cup\cQ^n_p contains $k$ edge-disjoint Hamilton cycles, for any fixed $k\in\mathbb{N}$. This result is part of a larger volume of work where we also prove the corresponding hitting time result for Hamiltonicity. In Chapter 3 we move to a non random setting. %to a deterministic one. %Instead of embedding a single Hamilton cycle our result concerns packing more general families of graphs into a fixed host graph. Rather than pack a small number of Hamilton cycles into a fixed host graph, our aim is to achieve optimally sized packings of more general families of graphs. More specifically, we provide a degree condition on a regular $n$-vertex graph $G$ which ensures the existence of a near optimal packing of any family $\mathcal H$ of bounded degree $n$-vertex $k$-chromatic separable graphs into $G$. %In general, this degree condition is best possible. %In particular, this yields an approximate version of the tree packing conjecture %in the setting of regular host graphs $G$ of high degree. %Similarly, our result implies approximate versions of the Oberwolfach problem, %the Alspach problem and the existence of resolvable designs in the setting of %regular host graphs of high degree. In particular, this yields approximate versions of the the tree packing conjecture, the Oberwolfach problem, the Alspach problem and the existence of resolvable designs in the setting of regular host graphs of high degree

Substructures in large graphs

The first problem we address concerns Hamilton cycles. Suppose G is a large digraph in which every vertex has in- and outdegree at least |G|/2. We show that G contains every orientation of a Hamilton cycle except, possibly, the antidirected one. The antidirected case was settled by DeBiasio and Molla. Our result is best possible and improves on an approximate result by Häggkvist and Thomason. We then investigate the random greedy F-free process which was initially studied by Erdős, Suen and Winkler and by Spencer. This process greedily adds edges without creating a copy of F, terminating in a maximal F-free graph. We provide an upper bound on the number of hyperedges at the end of this process for a large class of hypergraphs. The remainder of this thesis focuses on F-decompositions, i.e., whether the edge set of a graph can be partitioned into copies of F. We obtain the best known bounds on the minimum degree which ensures a K$_r$-decomposition of an r-partite graph, with applications to Latin squares. Lastly, we find exact bounds on the minimum degree for a large graph to have a C$_2$$_k$-decomposition where k≠3. In both cases, we assume necessary divisibility conditions are satisfied