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Spatial random graphs capture several important properties of real-world networks. We prove quenched results for the continuous-space version of scale-free percolation introduced in [14]. This is an undirected inhomogeneous random graph whose vertices are given by a Poisson point process in $\mathbb{R}^d$. Each vertex is equipped with a random weight, and the probability that two vertices are connected by an edge depends on their weights and on their distance. Under suitable conditions on the parameters of the model, we show that, for almost all realizations of the point process, the degree distributions of all the nodes of the graph follow a power law with the same tail at infinity. We also show that the averaged clustering coefficient of the graph is self-averaging. In particular, it is almost surely equal to the annealed clustering coefficient of one point, which is a strictly positive quantity.
It is a fact simple to establish that the mixing time of the simple random walk on a d-regular graph $G_n$ with n vertices is asymptotically bounded from below by $\frac {d }{d-2 } \frac {\log n}{\log (d-1)}$. Such a bound is obtained by comparing the walk on $G_n$ to the walk on d-regular tree $\mathcal{T} _d$. If one can map another transitive graph $\mathcal{G} $ onto $G_n$, then we can improve the strategy by using a comparison with the random walk on $\mathcal{G} $ (instead of that of $\mathcal{T} _d$), and we obtain a lower bound of the form $\frac {1}{\mathfrak{h} }\log n$, where $\mathfrak{h} $ is the entropy rate associated with $\mathcal{G} $. We call this the entropic lower bound.
It was recently proved that in the case $\mathcal{G} =\mathcal{T} _d$, this entropic lower bound (in that case $\frac {d }{d-2 } \frac {\log n}{\log (d-1)}$) is sharp when graphs have minimal spectral radius and thus that in that case the random walk exhibits cutoff at the entropic time. In this article, we provide a generalisation of the result by providing a sufficient condition on the spectra of the random walks on $G_n$ under which the random walk exhibits cutoff at the entropic time. It applies notably to anisotropic random walks on random d-regular graphs and to random walks on random n-lifts of a base graph (including nonreversible walks).
Vertex-primitive self-complementary graphs were proved to be affine or in product action by Guralnick et al. [‘On orbital partitions and exceptionality of primitive permutation groups’, Trans. Amer. Math. Soc.356 (2004), 4857–4872]. The product action type is known in some sense. In this paper, we provide a generic construction for the affine case and several families of new self-complementary Cayley graphs are constructed.
A graph $\Gamma $ is called $(G, s)$-arc-transitive if $G \le \text{Aut} (\Gamma )$ is transitive on the set of vertices of $\Gamma $ and the set of s-arcs of $\Gamma $, where for an integer $s \ge 1$ an s-arc of $\Gamma $ is a sequence of $s+1$ vertices $(v_0,v_1,\ldots ,v_s)$ of $\Gamma $ such that $v_{i-1}$ and $v_i$ are adjacent for $1 \le i \le s$ and $v_{i-1}\ne v_{i+1}$ for $1 \le i \le s-1$. A graph $\Gamma $ is called 2-transitive if it is $(\text{Aut} (\Gamma ), 2)$-arc-transitive but not $(\text{Aut} (\Gamma ), 3)$-arc-transitive. A Cayley graph $\Gamma $ of a group G is called normal if G is normal in $\text{Aut} (\Gamma )$ and nonnormal otherwise. Fang et al. [‘On edge transitive Cayley graphs of valency four’, European J. Combin.25 (2004), 1103–1116] proved that if $\Gamma $ is a tetravalent 2-transitive Cayley graph of a finite simple group G, then either $\Gamma $ is normal or G is one of the groups $\text{PSL}_2(11)$, $\text{M} _{11}$, $\text{M} _{23}$ and $A_{11}$. However, it was unknown whether $\Gamma $ is normal when G is one of these four groups. We answer this question by proving that among these four groups only $\text{M} _{11}$ produces connected tetravalent 2-transitive nonnormal Cayley graphs. We prove further that there are exactly two such graphs which are nonisomorphic and both are determined in the paper. As a consequence, the automorphism group of any connected tetravalent 2-transitive Cayley graph of any finite simple group is determined.
Magnant and Martin conjectured that the vertex set of any d-regular graph G on n vertices can be partitioned into
$n / (d+1)$
paths (there exists a simple construction showing that this bound would be best possible). We prove this conjecture when
$d = \Omega(n)$
, improving a result of Han, who showed that in this range almost all vertices of G can be covered by
$n / (d+1) + 1$
vertex-disjoint paths. In fact our proof gives a partition of V(G) into cycles. We also show that, if
$d = \Omega(n)$
and G is bipartite, then V(G) can be partitioned into n/(2d) paths (this bound is tight for bipartite graphs).
A tight Hamilton cycle in a k-uniform hypergraph (k-graph) G is a cyclic ordering of the vertices of G such that every set of k consecutive vertices in the ordering forms an edge. Rödl, Ruciński and Szemerédi proved that for
$k\ge 3$
, every k-graph on n vertices with minimum codegree at least
$n/2+o(n)$
contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such k-graphs is
${\exp(n\ln n-\Theta(n))}$
. As a corollary, we obtain a similar estimate on the number of Hamilton
${\ell}$
-cycles in such k-graphs for all
${\ell\in\{0,\ldots,k-1\}}$
, which makes progress on a question of Ferber, Krivelevich and Sudakov.
For a group G, we define a graph $\Delta (G)$ by letting $G^{\scriptsize\#}=G{\setminus} \lbrace 1\rbrace $ be the set of vertices and by drawing an edge between distinct elements $x,y\in G^{\scriptsize\#}$ if and only if the subgroup $\langle x,y\rangle $ is cyclic. Recall that a Z-group is a group where every Sylow subgroup is cyclic. In this short note, we investigate $\Delta (G)$ for a Z-group G.
We prove that, for any $t \ge 3$, there exists a constant c = c(t) > 0 such that any d-regular n-vertex graph with the second largest eigenvalue in absolute value λ satisfying $\lambda \le c{d^{t - 1}}/{n^{t - 2}}$ contains vertex-disjoint copies of kt covering all but at most ${n^{1 - 1/(8{t^4})}}$ vertices. This provides further support for the conjecture of Krivelevich, Sudakov and Szábo (Combinatorica24 (2004), pp. 403–426) that (n, d, λ)-graphs with n ∈ 3ℕ and $\lambda \le c{d^2}/n$ for a suitably small absolute constant c > 0 contain triangle-factors. Our arguments combine tools from linear programming with probabilistic techniques, and apply them in a certain weighted setting. We expect this method will be applicable to other problems in the field.
The general position number of a connected graph is the cardinality of a largest set of vertices such that no three pairwise-distinct vertices from the set lie on a common shortest path. In this paper it is proved that the general position number is additive on the Cartesian product of two trees.
Let G be a graph of minimum degree at least k and let Gp be the random subgraph of G obtained by keeping each edge independently with probability p. We are interested in the size of the largest complete minor that Gp contains when p = (1 + ε)/k with ε > 0. We show that with high probability Gp contains a complete minor of order
$\tilde{\Omega}(\sqrt{k})$
, where the ~ hides a polylogarithmic factor. Furthermore, in the case where the order of G is also bounded above by a constant multiple of k, we show that this polylogarithmic term can be removed, giving a tight bound.
An important problem in modeling networks is how to generate a randomly sampled graph with given degrees. A popular model is the configuration model, a network with assigned degrees and random connections. The erased configuration model is obtained when self-loops and multiple edges in the configuration model are removed. We prove an upper bound for the number of such erased edges for regularly-varying degree distributions with infinite variance, and use this result to prove central limit theorems for Pearson’s correlation coefficient and the clustering coefficient in the erased configuration model. Our results explain the structural correlations in the erased configuration model and show that removing edges leads to different scaling of the clustering coefficient. We prove that for the rank-1 inhomogeneous random graph, another null model that creates scale-free simple networks, the results for Pearson’s correlation coefficient as well as for the clustering coefficient are similar to the results for the erased configuration model.
Considering a natural generalization of the Ruzsa–Szemerédi problem, we prove that for any fixed positive integers r, s with r < s, there are graphs on n vertices containing
$n^{r}e^{-O\left(\sqrt{\log{n}}\right)}=n^{r-o(1)}$
copies of Ks such that any Kr is contained in at most one Ks. We also give bounds for the generalized rainbow Turán problem ex (n, H, rainbow - F) when F is complete. In particular, we answer a question of Gerbner, Mészáros, Methuku and Palmer, showing that there are properly edge-coloured graphs on n vertices with
$n^{r-1-o(1)}$
copies of Kr such that no Kr is rainbow.
For a finite group G, let
$\Delta (G)$
denote the character graph built on the set of degrees of the irreducible complex characters of G. A perfect graph is a graph
$\Gamma $
in which the chromatic number of every induced subgraph
$\Delta $
of
$\Gamma $
equals the clique number of
$\Delta $
. We show that the character graph
$\Delta (G)$
of a finite group G is always a perfect graph. We also prove that the chromatic number of the complement of
$\Delta (G)$
is at most three.
The Erdős–Simonovits stability theorem states that for all ε > 0 there exists α > 0 such that if G is a Kr+1-free graph on n vertices with e(G) > ex(n, Kr+1)– α n2, then one can remove εn2 edges from G to obtain an r-partite graph. Füredi gave a short proof that one can choose α = ε. We give a bound for the relationship of α and ε which is asymptotically sharp as ε → 0.
Let I be a zero-dimensional ideal in the polynomial ring
$K[x_1,\ldots ,x_n]$
over a field K. We give a bound for the number of roots of I in
$K^n$
counted with combinatorial multiplicity. As a consequence, we give a proof of Alon’s combinatorial Nullstellensatz.
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(d1, …, dn) as n→ ∞. We also determine the expected number of spanning trees in this model. The range of degrees covered includes dj= λn + O(n1/2+ε) for some λ bounded away from 0 and 1.
The disjointness graph G = G(𝒮) of a set of segments 𝒮 in
${\mathbb{R}^d}$
,
$$d \ge 2$$
, is a graph whose vertex set is 𝒮 and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of G satisfies
$\chi (G) \le {(\omega (G))^4} + {(\omega (G))^3}$
, where ω(G) denotes the clique number of G. It follows that 𝒮 has Ω(n1/5) pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments.
We show that computing ω(G) and χ(G) for disjointness graphs of lines in space are NP-hard tasks. However, we can design efficient algorithms to compute proper colourings of G in which the number of colours satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free (ω(G) = 2), but whose chromatic numbers are arbitrarily large.
A perfect Kr-tiling in a graph G is a collection of vertex-disjoint copies of the clique Kr in G covering every vertex of G. The famous Hajnal–Szemerédi theorem determines the minimum degree threshold for forcing a perfect Kr-tiling in a graph G. The notion of discrepancy appears in many branches of mathematics. In the graph setting, one assigns the edges of a graph G labels from {‒1, 1}, and one seeks substructures F of G that have ‘high’ discrepancy (i.e. the sum of the labels of the edges in F is far from 0). In this paper we determine the minimum degree threshold for a graph to contain a perfect Kr-tiling of high discrepancy.
It is known that for Kn,n equipped with i.i.d. exp (1) edge costs, the minimum total cost of a perfect matching converges to
$\zeta(2)=\pi^2/6$
in probability. Similar convergence has been established for all edge cost distributions of pseudo-dimension
$q \geq 1$
. In this paper we extend those results to all real positive q, confirming the Mézard–Parisi conjecture in the last remaining applicable case.
We study and classify proper q-colourings of the ℤd lattice, identifying three regimes where different combinatorial behaviour holds. (1) When
$q\le d+1$
, there exist frozen colourings, that is, proper q-colourings of ℤd which cannot be modified on any finite subset. (2) We prove a strong list-colouring property which implies that, when
$q\ge d+2$
, any proper q-colouring of the boundary of a box of side length
$n \ge d+2$
can be extended to a proper q-colouring of the entire box. (3) When
$q\geq 2d+1$
, the latter holds for any
$n \ge 1$
. Consequently, we classify the space of proper q-colourings of the ℤd lattice by their mixing properties.