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We use an inequality of Sidorenko to show a general relation between local and global subgraph counts and degree moments for locally weakly convergent sequences of sparse random graphs. This yields an optimal criterion to check when the asymptotic behaviour of graph statistics, such as the clustering coefficient and assortativity, is determined by the local weak limit.
As an application we obtain new facts for several common models of sparse random intersection graphs where the local weak limit, as we see here, is a simple random clique tree corresponding to a certain two-type Galton–Watson branching process.
Given a graph
$H$
and a positive integer
$n$
, the Turán number
$\mathrm{ex}(n,H)$
is the maximum number of edges in an
$n$
-vertex graph that does not contain
$H$
as a subgraph. A real number
$r\in (1,2)$
is called a Turán exponent if there exists a bipartite graph
$H$
such that
$\mathrm{ex}(n,H)=\Theta (n^r)$
. A long-standing conjecture of Erdős and Simonovits states that
$1+\frac{p}{q}$
is a Turán exponent for all positive integers
$p$
and
$q$
with
$q\gt p$
.
In this paper, we show that
$1+\frac{p}{q}$
is a Turán exponent for all positive integers
$p$
and
$q$
with
$q \gt p^{2}$
. Our result also addresses a conjecture of Janzer [18].
Cooperative coordination in multi-agent systems has been a topic of interest in networked control theory in recent years. In contrast to cooperative agents, Byzantine agents in a network are capable to manipulate their data arbitrarily and send bad messages to neighbors, causing serious network security issues. This paper is concerned with resilient tracking consensus over a time-varying random directed graph, which consists of cooperative agents, Byzantine agents and a single leader. The objective of resilient tracking consensus is the convergence of cooperative agents to the leader in the presence of those deleterious Byzantine agents. We assume that the number and identity of the Byzantine agents are not known to cooperative agents, and the communication edges in the graph are dynamically randomly evolving. Based upon linear system analysis and a martingale convergence theorem, we design a linear discrete-time protocol to ensure tracking consensus almost surely in a purely distributed manner. Some numerical examples are provided to verify our theoretical results.
The clustered chromatic number of a class of graphs is the minimum integer
$k$
such that for some integer
$c$
every graph in the class is
$k$
-colourable with monochromatic components of size at most
$c$
. We determine the clustered chromatic number of any minor-closed class with bounded treedepth, and prove a best possible upper bound on the clustered chromatic number of any minor-closed class with bounded pathwidth. As a consequence, we determine the fractional clustered chromatic number of every minor-closed class.
We study the problem of finding pairwise vertex-disjoint triangles in the randomly perturbed graph model, which is the union of any
$n$
-vertex graph
$G$
satisfying a given minimum degree condition and the binomial random graph
$G(n,p)$
. We prove that asymptotically almost surely
$G \cup G(n,p)$
contains at least
$\min \{\delta (G), \lfloor n/3 \rfloor \}$
pairwise vertex-disjoint triangles, provided
$p \ge C \log n/n$
, where
$C$
is a large enough constant. This is a perturbed version of an old result of Dirac.
Our result is asymptotically optimal and answers a question of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516] in a strong form. We also prove a stability version of our result, which in the case of pairwise vertex-disjoint triangles extends a result of Han, Morris, and Treglown [RSA, 2021, no. 3, 480–516]. Together with a result of Balogh, Treglown, and Wagner [CPC, 2019, no. 2, 159–176], this fully resolves the existence of triangle factors in randomly perturbed graphs.
We believe that the methods introduced in this paper are useful for a variety of related problems: we discuss possible generalisations to clique factors, cycle factors, and
$2$
-universality.
In this paper we define a family of preferential attachment models for random graphs with fitness in the following way: independently for each node, at each time step a random fitness is drawn according to the position of a moving average process with positive increments. We will define two regimes in which our graph reproduces some features of two well-known preferential attachment models: the Bianconi–Barabási and Barabási–Albert models. We will discuss a few conjectures on these models, including the convergence of the degree sequence and the appearance of Bose–Einstein condensation in the network when the drift of the fitness process has order comparable to the graph size.
A handlebody link is a union of handlebodies of positive genus embedded in 3-space, which generalises the notion of links in classical knot theory. In this paper, we consider handlebody links with a genus two handlebody and
$n-1$
solid tori,
$n>1$
. Our main result is the classification of such handlebody links with six crossings or less, up to ambient isotopy.
Empirical studies (e.g. Jiang et al. (2015) and Mislove et al. (2007)) show that online social networks have not only in- and out-degree distributions with Pareto-like tails, but also a high proportion of reciprocal edges. A classical directed preferential attachment (PA) model generates in- and out-degree distributions with power-law tails, but the theoretical properties of the reciprocity feature in this model have not yet been studied. We derive asymptotic results on the number of reciprocal edges between two fixed nodes, as well as the proportion of reciprocal edges in the entire PA network. We see that with certain choices of parameters, the proportion of reciprocal edges in a directed PA network is close to 0, which differs from the empirical observation. This points out one potential problem of fitting a classical PA model to a given network dataset with high reciprocity, and indicates that alternative models need to be considered.
In 1975 Bollobás, Erdős, and Szemerédi asked the following question: given positive integers
$n, t, r$
with
$2\le t\le r-1$
, what is the largest minimum degree
$\delta (G)$
among all
$r$
-partite graphs
$G$
with parts of size
$n$
and which do not contain a copy of
$K_{t+1}$
? The
$r=t+1$
case has attracted a lot of attention and was fully resolved by Haxell and Szabó, and Szabó and Tardos in 2006. In this article, we investigate the
$r\gt t+1$
case of the problem, which has remained dormant for over 40 years. We resolve the problem exactly in the case when
$r \equiv -1 \pmod{t}$
, and up to an additive constant for many other cases, including when
$r \geq (3t-1)(t-1)$
. Our approach utilizes a connection to the related problem of determining the maximum of the minimum degrees among the family of balanced
$r$
-partite
$rn$
-vertex graphs of chromatic number at most
$t$
.
The hyperbolic random geometric graph was introduced by Krioukov et al. (Phys. Rev. E82, 2010). Among many equivalent models for the hyperbolic space, we study the d-dimensional Poincaré ball (
$d\ge 2$
), with a general connectivity radius. While many phase transitions are known for the expectation asymptotics of certain subgraph counts, very little is known about the second-order results. Two of the distinguishing characteristics of geometric graphs on the hyperbolic space are the presence of tree-like hierarchical structures and the power-law behaviour of the degree distribution. We aim to reveal such characteristics in detail by investigating the behaviour of sub-tree counts. We show multiple phase transitions for expectation and variance in the resulting hyperbolic geometric graph. In particular, the expectation and variance of the sub-tree counts exhibit an intricate dependence on the degree sequence of the tree under consideration. Additionally, unlike the thermodynamic regime of the Euclidean random geometric graph, the expectation and variance may exhibit different growth rates, which is indicative of power-law behaviour. Finally, we also prove a normal approximation for sub-tree counts using the Malliavin–Stein method of Last et al. (Prob. Theory Relat. Fields165, 2016), along with the Palm calculus for Poisson point processes.
A hypergraph
$\mathcal{F}$
is non-trivial intersecting if every pair of edges in it have a nonempty intersection, but no vertex is contained in all edges of
$\mathcal{F}$
. Mubayi and Verstraëte showed that for every
$k \ge d+1 \ge 3$
and
$n \ge (d+1)k/d$
every
$k$
-graph
$\mathcal{H}$
on
$n$
vertices without a non-trivial intersecting subgraph of size
$d+1$
contains at most
$\binom{n-1}{k-1}$
edges. They conjectured that the same conclusion holds for all
$d \ge k \ge 4$
and sufficiently large
$n$
. We confirm their conjecture by proving a stronger statement.
They also conjectured that for
$m \ge 4$
and sufficiently large
$n$
the maximum size of a
$3$
-graph on
$n$
vertices without a non-trivial intersecting subgraph of size
$3m+1$
is achieved by certain Steiner triple systems. We give a construction with more edges showing that their conjecture is not true in general.
A conjecture of Alon, Krivelevich and Sudakov states that, for any graph
$F$
, there is a constant
$c_F \gt 0$
such that if
$G$
is an
$F$
-free graph of maximum degree
$\Delta$
, then
$\chi\!(G) \leqslant c_F \Delta/ \log\!\Delta$
. Alon, Krivelevich and Sudakov verified this conjecture for a class of graphs
$F$
that includes all bipartite graphs. Moreover, it follows from recent work by Davies, Kang, Pirot and Sereni that if
$G$
is
$K_{t,t}$
-free, then
$\chi\!(G) \leqslant (t + o(1)) \Delta/ \log\!\Delta$
as
$\Delta \to \infty$
. We improve this bound to
$(1+o(1)) \Delta/\log\!\Delta$
, making the constant factor independent of
$t$
. We further extend our result to the DP-colouring setting (also known as correspondence colouring), introduced by Dvořák and Postle.
Let
${\mathbb{G}(n_1,n_2,m)}$
be a uniformly random m-edge subgraph of the complete bipartite graph
${K_{n_1,n_2}}$
with bipartition
$(V_1, V_2)$
, where
$n_i = |V_i|$
,
$i=1,2$
. Given a real number
$p \in [0,1]$
such that
$d_1 \,{:\!=}\, pn_2$
and
$d_2 \,{:\!=}\, pn_1$
are integers, let
$\mathbb{R}(n_1,n_2,p)$
be a random subgraph of
${K_{n_1,n_2}}$
with every vertex
$v \in V_i$
of degree
$d_i$
,
$i = 1, 2$
. In this paper we determine sufficient conditions on
$n_1,n_2,p$
and m under which one can embed
${\mathbb{G}(n_1,n_2,m)}$
into
$\mathbb{R}(n_1,n_2,p)$
and vice versa with probability tending to 1. In particular, in the balanced case
$n_1=n_2$
, we show that if
$p\gg\log n/n$
and
$1 - p \gg \left(\log n/n \right)^{1/4}$
, then for some
$m\sim pn^2$
, asymptotically almost surely one can embed
${\mathbb{G}(n_1,n_2,m)}$
into
$\mathbb{R}(n_1,n_2,p)$
, while for
$p\gg\left(\log^{3} n/n\right)^{1/4}$
and
$1-p\gg\log n/n$
the opposite embedding holds. As an extension, we confirm the Kim–Vu Sandwich Conjecture for degrees growing faster than
$(n \log n)^{3/4}$
.
Hadwiger’s conjecture asserts that every graph without a
$K_t$
-minor is
$(t-1)$
-colourable. It is known that the exact version of Hadwiger’s conjecture does not extend to list colouring, but it has been conjectured by Kawarabayashi and Mohar (2007) that there exists a constant
$c$
such that every graph with no
$K_t$
-minor has list chromatic number at most
$ct$
. More specifically, they also conjectured that this holds for
$c=\frac{3}{2}$
.
Refuting the latter conjecture, we show that the maximum list chromatic number of graphs with no
$K_t$
-minor is at least
$(2-o(1))t$
, and hence
$c \ge 2$
in the above conjecture is necessary. This improves the previous best lower bound by Barát, Joret and Wood (2011), who proved that
$c \ge \frac{4}{3}$
. Our lower-bound examples are obtained via the probabilistic method.
For a subgraph
$G$
of the blow-up of a graph
$F$
, we let
$\delta ^*(G)$
be the smallest minimum degree over all of the bipartite subgraphs of
$G$
induced by pairs of parts that correspond to edges of
$F$
. Johansson proved that if
$G$
is a spanning subgraph of the blow-up of
$C_3$
with parts of size
$n$
and
$\delta ^*(G) \ge \frac{2}{3}n + \sqrt{n}$
, then
$G$
contains
$n$
vertex disjoint triangles, and presented the following conjecture of Häggkvist. If
$G$
is a spanning subgraph of the blow-up of
$C_k$
with parts of size
$n$
and
$\delta ^*(G) \ge \left(1 + \frac 1k\right)\frac n2 + 1$
, then
$G$
contains
$n$
vertex disjoint copies of
$C_k$
such that each
$C_k$
intersects each of the
$k$
parts exactly once. A similar conjecture was also made by Fischer and the case
$k=3$
was proved for large
$n$
by Magyar and Martin.
In this paper, we prove the conjecture of Häggkvist asymptotically. We also pose a conjecture which generalises this result by allowing the minimum degree conditions in each bipartite subgraph induced by pairs of parts of
$G$
to vary. We support this new conjecture by proving the triangle case. This result generalises Johannson’s result asymptotically.
A graph
$H$
is common if the number of monochromatic copies of
$H$
in a 2-edge-colouring of the complete graph
$K_n$
is asymptotically minimised by the random colouring. Burr and Rosta, extending a famous conjecture of Erdős, conjectured that every graph is common. The conjectures of Erdős and of Burr and Rosta were disproved by Thomason and by Sidorenko, respectively, in the late 1980s. Collecting new examples of common graphs had not seen much progress since then, although very recently a few more graphs were verified to be common by the flag algebra method or the recent progress on Sidorenko’s conjecture. Our contribution here is to provide several new classes of tripartite common graphs. The first example is the class of so-called triangle trees, which generalises two theorems by Sidorenko and answers a question of Jagger, Šťovíček, and Thomason from 1996. We also prove that, somewhat surprisingly, given any tree
$T$
, there exists a triangle tree such that the graph obtained by adding
$T$
as a pendant tree is still common. Furthermore, we show that adding arbitrarily many apex vertices to any connected bipartite graph on at most
$5$
vertices yields a common graph.
We show that the diameter of a uniformly drawn spanning tree of a simple connected graph on n vertices with minimal degree linear in n is typically of order
$\sqrt{n}$
. A byproduct of our proof, which is of independent interest, is that on such graphs the Cheeger constant and the spectral gap are comparable.
The random-cluster model is a unifying framework for studying random graphs, spin systems and electrical networks that plays a fundamental role in designing efficient Markov Chain Monte Carlo (MCMC) sampling algorithms for the classical ferromagnetic Ising and Potts models. In this paper, we study a natural non-local Markov chain known as the Chayes–Machta (CM) dynamics for the mean-field case of the random-cluster model, where the underlying graph is the complete graph on n vertices. The random-cluster model is parametrised by an edge probability p and a cluster weight q. Our focus is on the critical regime:
$p = p_c(q)$
and
$q \in (1,2)$
, where
$p_c(q)$
is the threshold corresponding to the order–disorder phase transition of the model. We show that the mixing time of the CM dynamics is
$O({\log}\ n \cdot \log \log n)$
in this parameter regime, which reveals that the dynamics does not undergo an exponential slowdown at criticality, a surprising fact that had been predicted (but not proved) by statistical physicists. This also provides a nearly optimal bound (up to the
$\log\log n$
factor) for the mixing time of the mean-field CM dynamics in the only regime of parameters where no non-trivial bound was previously known. Our proof consists of a multi-phased coupling argument that combines several key ingredients, including a new local limit theorem, a precise bound on the maximum of symmetric random walks with varying step sizes and tailored estimates for critical random graphs. In addition, we derive an improved comparison inequality between the mixing time of the CM dynamics and that of the local Glauber dynamics on general graphs; this results in better mixing time bounds for the local dynamics in the mean-field setting.