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 which guarantee a perfect F-packing.
The Regularity lemma of Szemeredi and the Blow-up lemma of Komlos, Sarkozy and Szemeredi have proved to be powerful tools in attacking such problems and quite recently, several long-standing problems and conjectures in the area have been solved using these. In this survey, we give an outline of recent progress (with our main emphasis on F-packings, Hamiltonicity problems and tree embeddings) and describe some of the methods involved.
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…
Given graphs $G$ and $H$, we propose a method to implicitly enumerate topological-minor-embeddings of $H$ in $G$ using decision diagrams. We show a useful application of our method to enumerating subgraphs characterized by forbidden topological minors, that is, planar, outerplanar, series-parallel, and cactus subgraphs. Computational experiments show that our method can find all planar subgraphs in a given graph at most five orders of magnitude faster than a naive backtracking-based method.
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.
For any graph G with n edges, the spanning subgraphs and the orientations of G are both counted by the evaluation T_G(2,2)=2^n of its Tutte polynomial. We define a bijection $Φ$ between spanning subgraphs and orientations and explore its enumerative consequences regarding the Tutte polynomial. The bijection $Φ$ is closely related to a recent characterization of the Tutte polynomial relying on a combinatorial embedding of the graph G, that is, on a choice of cyclic order of the edges around each vertex. Among other results, we obtain a combinatorial interpretation for each of the evaluations T_…
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…
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…
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…
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].
It follows from the work of Tait and the Four-Color-Theorem that a planar cubic graph is 3-edge-colorable if and only if it contains no bridge. We consider the question of which planar graphs are subgraphs of planar cubic bridgeless graphs, and hence 3-edge-colorable. We provide an efficient recognition algorithm that given an $n$-vertex planar graph, augments this graph in $O(n^2)$ steps to a planar cubic bridgeless supergraph, or decides that no such augmentation is possible. The main tools involve the Generalized Antifactor-problem for the fixed embedding case, and SPQR-trees for the variab…
