Related papers: Long paths in random Apollonian networks

Given integers $k,j$ with $1\le j \le k-1$, we consider the length of the longest $j$-tight path in the binomial random $k$-uniform hypergraph $H^k(n,p)$. We show that this length undergoes a phase transition from logarithmic length to…

Given a graph $G$ and a bijection $f : E(G)\rightarrow \{1, 2, \ldots,e(G)\}$, we say that a trail/path in $G$ is $f$-\emph{increasing} if the labels of consecutive edges of this trail/path form an increasing sequence. More than 40 years…

In this paper, we study the distribution of distances in random Apollonian network structures (RANS), a family of graphs which has a one-to-one correspondence with planar ternary trees. Using multivariate generating functions that express…

In a model of a connected network on random points in the plane, one expects that the mean length of the shortest route between vertices at distance $r$ apart should grow only as $O(r)$ as $r \to \infty$, but this is not always easy to…

We prove almost tight bounds on the length of paths in $2$-edge-connected cubic graphs. Concretely, we show that (i) every $2$-edge-connected cubic graph of size $n$ has a path of length $\Omega\left(\frac{\log^2{n}}{\log{\log{n}}}\right)$,…

We revisit several maximization problems for geometric networks design under the non-crossing constraint, first studied by Alon, Rajagopalan and Suri (ACM Symposium on Computational Geometry, 1993). Given a set of $n$ points in the plane in…

We introduce a family of complex networks that interpolates between the Apollonian network and its binary version, recently introduced in [Phys. Rev. E \textbf{107}, 024305 (2023)], via random removal of nodes. The dilution process allows…

For a given finite graph $G$ of minimum degree at least $k$, let $G_{p}$ be a random subgraph of $G$ obtained by taking each edge independently with probability $p$. We prove that (i) if $p \ge \omega/k$ for a function $\omega=\omega(k)$…

Let $P$ be a set of $2n$ points in convex position, such that $n$ points are colored red and $n$ points are colored blue. A non-crossing alternating path on $P$ of length $\ell$ is a sequence $p_1, \dots, p_\ell$ of $\ell$ points from $P$…

We present a family of networks, expanded deterministic Apollonian networks, which are a generalization of the Apollonian networks and are simultaneously scale-free, small-world, and highly clustered. We introduce a labeling of their…

Analytic solution for the average path length in a large class of random graphs is found. We apply the approach to classical random graphs of Erd\"{o}s and R\'{e}nyi (ER) and to scale-free networks of Barab\'{a}si and Albert (BA). In both…

In a uniform random recursive k-dag, there is a root, 0, and each node in turn, from 1 to n, chooses k uniform random parents from among the nodes of smaller index. If S_n is the shortest path distance from node n to the root, then we…

In this article, we propose a simple rule that generates scale-free networks with very large clustering coefficient and very small average distance. These networks are called {\bf Random Apollonian Networks}(RAN) as they can be considered…

We give a randomized algorithm that determines if a given graph has a simple path of length at least k in O(2^k poly(n,k)) time.

We determine the probability thresholds for the existence of monotone paths, of finite and infinite length, in random oriented graphs with vertex set $\mathbb N^{[k]}$, the set of all increasing $k$-tuples in $\mathbb N$. These graphs…

In graph theory, the longest path problem is the problem of finding a simple path of maximum length in a given graph. For some small classes of graphs, the problem can be solved in polynomial time [2, 4], but it remains NP-hard on general…

In this work we analyze basic properties of Random Apollonian Networks \cite{zhang,zhou}, a popular stochastic model which generates planar graphs with power law properties. Specifically, let $k$ be a constant and $\Delta_1 \geq \Delta_2…

We consider a directed graph on the 2-dimensional integer lattice, placing a directed edge from vertex $(i_1,i_2)$ to $(j_1,j_2)$, whenever $i_1 \le j_1$, $i_2 \le j_2$, with probability $p$, independently for each such pair of vertices.…

We provide a new lower bound on the length of the longest cycle of the binomial random graph $G(n,(1+\epsilon)/n)$ that holds w.h.p. for all $\epsilon=\epsilon(n)$ such that $\epsilon^3n\to \infty$. In the case $\epsilon\leq \epsilon_0$ for…

A \emph{random temporal graph} is an Erd\H{o}s-R\'enyi random graph $G(n,p)$, together with a random ordering of its edges. A path in the graph is called \emph{increasing} if the edges on the path appear in increasing order. A set $S$ of…