Related papers: Large cycles in random generalized Johnson graphs
We prove that a random cubic graph almost surely is not homomorphic to a cycle of size 7. This implies that there exist cubic graphs of arbitrarily high girth with no homomorphisms to the cycle of size 7.
We analyse the scaling limit of the sizes of the largest components of the Random Intersection Graph $G(n,m,p)$ close to the critical point $p=\frac{1}{\sqrt{nm}}$, when the numbers $n$ of individuals and $m$ of communities have different…
In this paper we extend counting of traversing Hamiltonian cycles from 2-tiled graphs to generalized tiled graphs. We further show that, for a fixed finite set of tiles, counting traversing Hamiltonian cycles can be done in linear time with…
Two new sufficient conditions for generalized cycles (including Hamilton and dominating cycles as special cases) in an arbitrary k-connected graph (k=1,2,...) are derived, which prove the truth of Bondy's (1980) famous conjecture for some…
In this short note, we give a new sufficient condition for the existence of long cycles in graphs involving Fan-type degree condition and neighborhood intersection.
This survey concerns regular graphs that are extremal with respect to the number of independent sets, and more generally, graph homomorphisms. More precisely, in the family of of $d$-regular graphs, which graph $G$ maximizes/minimizes the…
Inspired by a concept in comparative genomics, we investigate properties of randomly chosen members of G_1(m,n,t), the set of bipartite graphs with $m$ left vertices, n right vertices, t edges, and each vertex of degree at least one. We…
Consider n unit intervals, say [1,2], [3,4], ..., [2n-1,2n]. Identify their endpoints in pairs at random, with all (2n-1)!! = (2n-1) (2n-3) ... 3 1 pairings being equally likely. The result is a collection of cycles of various lengths, and…
A version of ``preferential attachment'' random graphs, corresponding to linear ``weights'' with random ``edge additions,'' which generalizes some previously considered models, is studied. This graph model is embedded in a continuous-time…
A single permutation, seen as union of disjoint cycles, represents a regular graph of degree two. Consider $d$ many independent random permutations and superimpose their graph structures. It is a common model of a random regular (multi-)…
Let $\mathcal{G}(n,r,s)$ denote a uniformly random $r$-regular $s$-uniform hypergraph on $n$ vertices, where $s$ is a fixed constant and $r=r(n)$ may grow with $n$. An $\ell$-overlapping Hamilton cycle is a Hamilton cycle in which…
In stochastic geometry there are several instances of threshold phenomena in high dimensions: the behavior of a limit of some expectation changes abruptly when some parameter passes through a critical value. This note continues the…
We introduce a solvable model of randomly growing systems consisting of many independent subunits. Scaling relations and growth rate distributions in the limit of infinite subunits are analysed theoretically. Various types of scaling…
In this paper we show that $e/n$ is the sharp threshold for the existence of tight Hamilton cycles in random $k$-uniform hypergraphs, for all $k\ge 4$. When $k=3$ we show that $1/n$ is an asymptotic threshold. We also determine thresholds…
Each connected component of a mapping $\{1,2,...,n\}\rightarrow\{1,2,...,n\}$ contains a unique cycle. The largest such component can be studied probabilistically via either a delay differential equation or an inverse Laplace transform. The…
A random geometric graph $G(\mathcal{X}_n, r_n)$ is formed by taking a binomial process $\mathcal{X}_n$ as the set of vertices and joining any two distinct points with an edge if they lie within distance $r_n$ of each other. We investigate…
We consider a model for random hypergraphs with identifiability, an analogue of connectedness. This model has a phase transition in the proportion of identifiable vertices when the underlying random graph becomes critical. The phase…
We introduce a model for a growing random graph based on simultaneous reproduction of the vertices. The model can be thought of as a generalisation of the reproducing graphs of Southwell and Cannings and Bonato et al to allow for a random…
For uniform random permutations conditioned to have no long cycles, we prove that the total number of cycles satisfies a central limit theorem. Under additional assumptions on the asymptotic behavior of the set of allowed cycle lengths, we…
Let $L_{c,n}$ denote the size of the longest cycle in $G(n,{c}/{n})$, $c>1$ constant. We show that there exists a continuous function $f(c)$ such that $ L_{c,n}/n \to f(c)$ a.s. for $c\geq 20$, thus extending a result of the author and…