Given a family ${\cal F}$ of graphs, and a positive integer $n$, the Turán number $ex(n,{\cal F})$ of ${\cal F}$ is the maximum number of edges in an $n$-vertex graph that does not contain any member of ${\cal F}$ as a subgraph. The order of a graph is the number of vertices in it. In this paper, we study the Turán number of the family of graphs with bounded order and high average degree. For every real $d\geq 2$ and positive integer $m\geq 2$, let ${\cal F}_{d,m}$ denote the family of graphs on at most $m$ vertices that have average degree at least $d$. It follows from the Erdős-Rényi bound that $ex(n,{\cal F}_{d,m})=Ω(n^{2-\frac{2}{d}+\frac{c}{dm}})$, for some positive constant $c$. Verstraëte asked if it is true that for each fixed $d$ there exists a function $ε_d(m)$ that tends to $0$ as $m\to \infty$ such that $ex(n,{\cal F}_{d,m})=O(n^{2-\frac{2}{d}+ε_d(m)})$. We answer Verstraëte's question in the affirmative whenever $d$ is an integer. We also prove an extension of the cube theorem on the Turán number of the cube $Q_3$, which partially answers a question of Pinchasi and Sharir.
We prove that every bipartite graph of sufficiently large average degree has either a $K_{t,t}$-subgraph or an induced subgraph of average degree at least $t$ and girth at least $6$. We conjecture that "$6$" can be replaced by "$k$", which strengthens a conjecture of Thomassen. In support of this conjecture, we show that it holds for regular graphs.
This exposition contains a short and streamlined proof of the recent result of Kwan, Letzter, Sudakov and Tran that every triangle-free graph with minimum degree $d$ contains an induced bipartite subgraph with average degree $Ω(\ln d/\ln\ln d)$.
We determine how large r-partite graphs can be found in r-uniform graphs with n vertices and Cn^r edges, where C is a slowly decreasing function of n. This refines results of Erdos from 1964.
Kühn, Osthus and Taraz showed that for each γ>0 there exists C such that any n-vertex graph with minimum degree γn contains a planar subgraph with at least 2n-C edges. We find the optimum value of C for all γ<1/2 and sufficiently large n.
The problem of finding dense induced bipartite subgraphs in $H$-free graphs has a long history, and was posed 30 years ago by Erdős, Faudree, Pach and Spencer. In this paper, we obtain several results in this direction. First we prove that any $K_t$-free graph with minimum degree at least $d$ contains an induced bipartite subgraph of minimum degree at least $c_t \log d/\log \log d$, confirming (asymptotically) several conjectures by Esperet, Kang and Thomassé. Complementing this result, we further obtain optimal bounds for this problem in the case of dense triangle-free graphs, and we also ans…
Consider the triangle-free graph process, which starts from the empty graph on $n$ vertices and a random ordering of the possible ${n \choose 2}$ edges; the edges are added in this ordering provided the graph remains triangle free. We will show that there exists a constant $c$ such that no copy of any fixed finite triangle-free graph on $k$ vertices with at least $ck$ edges asymptotically almost surely appears in the triangle-free graph process.
We consider the structure of $H$-free subgraphs of graphs with high minimal degree. We prove that for every $k>m$ there exists an $ε:=ε(k,m)>0$ so that the following holds. For every graph $H$ with chromatic number $k$ from which one can delete an edge and reduce the chromatic number, and for every graph $G$ on $n>n_0(H)$ vertices in which all degrees are at least $(1-ε)n$, any subgraph of $G$ which is $H$-free and contains the maximum number of copies of the complete graph $K_m$ is $(k-1)$-colorable. We also consider several extensions for the case of a general forbidden graph $H$ of a given …
In this note, we show that for any $m \in \{1,2, \dots , q +1 \}$, if $G$ is a polarity graph of a projective plane of order $q$ that has an oval, then $G$ contains a subgraph on $m + \binom{m}{2}$ vertices with $m^2+\frac{m^4}{8q} - O ( \frac{m^4}{q^{3/2} } +m )$ edges. As an application, we give the best known lower bounds on the Turán number $\mathrm{ex}(n, C_4)$ for certain values of $n$. In particular, we disprove a conjecture of Abreu, Balbuena, and Labbate concerning $\mathrm{ex}(q^2-q-2, C_4)$ where $q$ is a power of $2$.
Motzkin and Straus established a remarkable connection between the maximum clique and the Lagrangian of a graph in 1965. This connection and its extensions were successfully employed in optimization to provide heuristics for the maximum clique number in graphs. It is useful in practice if similar results hold for hypergraphs. In this paper, we provide upper bounds on the Lagrangian of a hypergraph containing dense subgraphs when the number of edges of the hypergraph is in certain ranges. These results support a pair of conjectures introduced by Y. Peng and C. Zhao (2012) and extend a result of…
Given $p \in (0,1)$, we let $Q_p= Q_p^d$ be the random subgraph of the $d$-dimensional hypercube $Q^d$ where edges are present independently with probability $p$. It is well known that, as $d \rightarrow \infty$, if $p>\frac12$ then with high probability $Q_p$ is connected; and if $p<\frac12$ then with high probability $Q_p$ consists of one giant component together with many smaller components which form the `fragment'. Here we fix $p \in (0,\frac12)$, and investigate the fragment, and how it sits inside the hypercube. In particular we give asymptotic estimates for the mean numbers of componen…
We prove two estimates for the expectation of the exponential of a complex function of a random permutation or subset. Using this theory, we find asymptotic expressions for the expected number of copies and induced copies of a given graph in a uniformly random graph with degree sequence $(d_1,\ldots,d_n)$ as $n \rightarrow \infty$. We also determine the expected number of spanning trees in this model. The range of degrees covered includes $d_j = λn + O(n^{1/2+\varepsilon})$ for some $λ$ bounded away from $0$ and $1$.
We consider sufficient conditions for the existence of $k$-th powers of Hamiltonian cycles in $n$-vertex graphs $G$ with minimum degree $μn$ for arbitrarily small $μ>0$. About 20 years ago Komlós, Sarközy, and Szemerédi resolved the conjectures of Pósa and Seymour and obtained optimal minimum degree conditions for this problem by showing that $μ=\frac{k}{k+1}$ suffices for large $n$. For smaller values of $μ$ the given graph $G$ must satisfy additional assumptions. We show that inducing subgraphs of density $d>0$ on linear subsets of vertices and being inseparable, in the sense that every cut …
We consider the following random model for edge-colored graphs. A graph $G$ on $n$ vertices is fixed, and a random subgraph $G_p$ is chosen by letting each edge of $G$ remain independently with probability $p$. Then, each edge of $G_p$ is colored uniformly at random from the set $[n-1]$. A result of Frieze and McKay (Random Structures and Algorithms, 1994) implies that when $G = K_n$ and $p = (2 + ε) \frac{\log n}{n}$ for some constant $ε> 0$, then $G_p$ almost surely contains a rainbow spanning tree. In this paper, we show that if $G$ is a $d$-regular $Ω(n)$-edge-connected graph, then when $p…
We consider the Densest-Subgraph problem, where a graph and an integer k is given and we search for a subgraph on exactly k vertices that induces the maximum number of edges. We prove that this problem is NP-hard even when the input graph has maximum degree three.
What conditions ensure that a graph G contains some given spanning subgraph H? The most famous examples of results of this kind are probably Dirac's theorem on Hamilton cycles and Tutte's theorem on perfect matchings. Perfect matchings are generalized by perfect F-packings, where instead of covering all the vertices of G by disjoint edges, we want to cover G by disjoint copies of a (small) graph F. It is unlikely that there is a characterization of all graphs G which contain a perfect F-packing, so as in the case of Dirac's theorem it makes sense to study conditions on the minimum degree of G …
Let d = (d1, d2, ..., dn) be a vector of non-negative integers with even sum. We prove some basic facts about the structure of a random graph with degree sequence d, including the probability of a given subgraph or induced subgraph. Although there are many results of this kind, they are restricted to the sparse case with only a few exceptions. Our focus is instead on the case where the average degree is approximately a constant fraction of n. Our approach is the multidimensional saddle-point method. This extends the enumerative work of McKay and Wormald (1990) and is analogous to the theory de…
We consider sequences of graphs and define various notions of convergence related to these sequences: ``left convergence'' defined in terms of the densities of homomorphisms from small graphs into the graphs of the sequence, and ``right convergence'' defined in terms of the densities of homomorphisms from the graphs of the sequence into small graphs; and convergence in a suitably defined metric. In Part I of this series, we show that left convergence is equivalent to convergence in metric, both for simple graphs, and for graphs with nodeweights and edgeweights. One of the main steps here is th…
Consider the random subgraph process on a base graph $G$ on $n$ vertices: a sequence $\lbrace G_t \rbrace _{t=0} ^{|E(G)|}$ of random subgraphs of $G$ obtained by choosing an ordering of the edges of $G$ uniformly at random, and by sequentially adding edges to $G_0$, the empty graph on the vertex set of $G$, according to the chosen ordering. We show that if $G$ has one of the following properties: 1. There is a positive constant $\varepsilon > 0$ such that $δ(G) \geq \left( \frac{1}{2} + \varepsilon \right) n$; 2. There are some constants $α, β>0$ such that every two disjoint subsets $U,W$ of …
The H-free process starts with the empty graph on n vertices and adds edges chosen uniformly at random, one at a time, subject to the condition that no copy of H is created, where H is some fixed graph. When H is strictly 2-balanced, we show that for some c,d>0, with high probability as $n \to \infty$, the final graph of the H-free process contains no subgraphs F on $v_F \leq n^{d}$ vertices with maximum density $\max_{J \subseteq F}\{e_J/v_J\} \geq c$. This extends and generalizes results of Gerke and Makai for the C_3-free process.
A graph $G$ is weakly $γ$-closed if every induced subgraph of $G$ contains one vertex $v$ such that for each non-neighbor $u$ of $v$ it holds that $|N(u)\cap N(v)|<γ$. The weak closure $γ(G)$ of a graph, recently introduced by Fox et al. [SIAM J. Comp. 2020], is the smallest number such that $G$ is weakly $γ$-closed. This graph parameter is never larger than the degeneracy (plus one) and can be significantly smaller. Extending the work of Fox et al. [SIAM J. Comp. 2020] on clique enumeration, we show that several problems related to finding dense subgraphs, such as the enumeration of bicliques…
Mader proved that every strongly $k$-connected $n$-vertex digraph contains a strongly $k$-connected spanning subgraph with at most $2kn - 2k^2$ edges, where the equality holds for the complete bipartite digraph ${DK}_{k,n-k}$. For dense strongly $k$-connected digraphs, this upper bound can be significantly improved. More precisely, we prove that every strongly $k$-connected $n$-vertex digraph $D$ contains a strongly $k$-connected spanning subgraph with at most $kn + 800k(k+\overlineΔ(D))$ edges, where $\overlineΔ(D)$ denotes the maximum degree of the complement of the underlying undirected gra…
We study a restricted form of list colouring, for which every pair of lists that correspond to adjacent vertices may not share more than one colour. The optimal list size such that a proper list colouring is always possible given this restriction, we call separation choosability. We show for bipartite graphs that separation choosability increases with (the logarithm of) the minimum degree. This strengthens results of Molloy and Thron and, partially, of Alon. One attempt to drop the bipartiteness assumption precipitates a natural class of Ramsey-type questions, of independent interest. For exam…
Let $G$ be a simple graph with maximum degree $Δ(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>Δ(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ on $n$ vertices with $Δ(G)>n/3$ has chromatic index $Δ(G)$ if and only if $G$ contains no overfull subgraph. Glock, Kühn and Osthus in 2016 showed that the conjecture is true for dense quasirandom graphs with even order, and they conjectured that the same should hold for such graphs with odd order. In this paper, we show that the conjecture of Glock, Kühn and Osthus is affirmative.
We show that if a sequence of dense graphs has the property that for every fixed graph F, the density of copies of F in these graphs tends to a limit, then there is a natural ``limit object'', namely a symmetric measurable 2-variable function on [0,1]. This limit object determines all the limits of subgraph densities. We also show that the graph parameters obtained as limits of subgraph densities can be characterized by ``reflection positivity'', semidefiniteness of an associated matrix. Conversely, every such function arises as a limit object. Along the lines we introduce a rather general mod…
Dense regions in networks are an indicator of interesting and unusual information. However, most existing methods only consider simple, undirected, unweighted networks. Complex networks in the real-world often have rich information though: edges are asymmetrical and nodes/edges have categorical and numerical attributes. Finding dense subgraphs in such networks in accordance with this rich information is an important problem with many applications. Furthermore, most existing algorithms ignore the higher-order relationships (i.e., motifs) among the nodes. Motifs are shown to be helpful for dense…
For a constant $γ\in[0,1]$ and a graph $G$, let $ω_γ(G)$ be the largest integer $k$ for which there exists a $k$-vertex subgraph of $G$ with at least $γ\binom{k}{2}$ edges. We show that if $0<p<γ<1$ then $ω_γ(G_{n,p})$ is concentrated on a set of two integers. More precisely, with $α(γ,p)=γ\log\fracγ{p}+(1-γ)\log\frac{1-γ}{1-p}$, we show that $ω_γ(G_{n,p})$ is one of the two integers closest to $\frac{2}{α(γ,p)}\big(\log n-\log\log n+\log\frac{eα(γ,p)}{2}\big)+\frac{1}{2}$, with high probability. While this situation parallels that of cliques in random graphs, a new technique is required to ha…
For a fixed graph $H$ and for arbitrarily large host graphs $G$, the number of homomorphisms from $H$ to $G$ and the number of subgraphs isomorphic to $H$ contained in $G$ have been extensively studied in extremal graph theory and graph limits theory when the host graphs are allowed to be dense. This paper addresses the case when the host graphs are robustly sparse and proves a general theorem that solves a number of open questions proposed since 1990s and strengthens a number of results in the literature. We prove that for any graph $H$ and any set ${\mathcal H}$ of homomorphisms from $H$ to …
We determine the asymptotic behavior of the maximum subgraph density of large random graphs with a prescribed degree sequence. The result applies in particular to the Erdős-Rényi model, where it settles a conjecture of Hajek [IEEE Trans. Inform. Theory 36 (1990) 1398-1414]. Our proof consists in extending the notion of balanced loads from finite graphs to their local weak limits, using unimodularity. This is a new illustration of the objective method described by Aldous and Steele [In Probability on Discrete Structures (2004) 1-72 Springer].
Let $H_n$ be a graph on $n$ vertices and let $\ber{H_n}$ denote the complement of $H_n$. Suppose that $Δ= Δ(n)$ is the maximum degree of $\ber{H_n}$. We analyse three algorithms for sampling $d$-regular subgraphs ($d$-factors) of $H_n$. This is equivalent to uniformly sampling $d$-regular graphs which avoid a set $E(\ber{H_n})$ of forbidden edges. Here $d=d(n)$ is a positive integer which may depend on $n$. Two of these algorithms produce a uniformly random $d$-factor of $H_n$ in expected runtime which is linear in $n$ and low-degree polynomial in $d$ and $Δ$. The first algorithm applies when …
Let S and T be vectors of positive integers with the same sum. We study the uniform distribution on the space of simple bipartite graphs with degree sequence S in one part and T in the other; equivalently, binary matrices with row sums S and column sums T. In particular, we find precise formulae for the probabilities that a given bipartite graph is edge-disjoint from, a subgraph of, or an induced subgraph of a random graph in the class. We also give similar formulae for the uniform distribution on the set of simple directed graphs with out-degrees S and in-degrees T. In each case, the graphs o…
