Related papers: Finding long cycles in graphs
We discuss the existence of Hamilton cycles in the random graph $G_{n,p}$ where there are restrictions caused by (i) coloring sequences, (ii) a subset of vertices must occur in a specific order and (iii) there is a bound on the number of…
Markov chains are a convenient means of generating realizations of networks, since they require little more than a procedure for rewiring edges. If a rewiring procedure exists for generating new graphs with specified statistical properties,…
The subdivided double construction on 4-regular graphs was used by Poto\v{c}nik and Wilson to explore semi-symmetric (edge-transitive but not vertex-transitive) graphs, and can be used to construct every semi-symmetric 4-regular graph that…
In this paper, a polynomial time algorithm for finding the set of all cyclic subsets in a graph is presented. The concept of cyclic subsets has already been introduced in an earlier paper. The algorithm finds cyclic subsets in a graph G by…
Counting short cycles in bipartite graphs is a fundamental problem of interest in the analysis and design of low-density parity-check (LDPC) codes. The vast majority of research in this area is focused on algorithmic techniques. Most…
Computation of the probability that a random graph is connected is a challenging problem, so it is natural to turn to approximations such as Monte Carlo methods. We describe sequential importance resampling and splitting algorithms for the…
We generalize the Hamiltonian Monte Carlo algorithm with a stack of neural network layers and evaluate its ability to sample from different topologies in a two dimensional lattice gauge theory. We demonstrate that our model is able to…
We present the first optimal algorithm for the classical problem of listing all the cycles in an undirected graph. We exploit their properties so that the total cost is the time taken to read the input graph plus the time to list the…
A well-known result due to Chvat\'al and Erd\H{o}s (1972) asserts that, if a graph $G$ satisfies $\kappa(G) \ge \alpha(G)$, where $\kappa(G)$ is the vertex-connectivity of $G$, then $G$ has a Hamilton cycle. We prove a similar result…
We consider the parameterised $k,e$-Long Cycle problem, in which you are given an $n$-vertex undirected graph $G$, a specified edge $e$ in $G$, and a positive integer $k$, and are asked to decide if the graph $G$ has a simple cycle through…
Sampling from combinatorial families can be difficult. However, complicated families can often be embedded within larger, simpler ones, for which easy sampling algorithms are known. We take advantage of such a relationship to describe a…
We consider the problem of enumerating all instances of a given pattern graph in a large data graph. Our focus is on determining the input/output (I/O) complexity of this problem. Let $E$ be the number of edges in the data graph, $k=O(1)$…
Let ${\cal{C}}_1$ be the set of fundamental cycles of breadth-first-search trees in a graph $G$ and ${\cal{C}}_2$ the set of the sums of two cycles in ${\cal{C}}_1$. Then we show that $(1) {\cal{C}}={\cal{C}}_1\bigcup{\cal{C}}_2$ contains a…
Structural properties of evolving random graphs are investigated. Treating linking as a dynamic aggregation process, rate equations for the distribution of node to node distances (paths) and of cycles are formulated and solved analytically.…
In this paper we consider the existence of Hamilton cycles in the random graph $G=G_{n,m}^{\delta\geq 3}$. This a random graph chosen uniformly from the set of graphs with vertex set $[n]$, $m$ edges and minimum degree at least 3. Our…
A class of Monte Carlo algorithms which incorporate absorbing Markov chains is presented. In a particular limit, the lowest-order of these algorithms reduces to the $n$-fold way algorithm. These algorithms are applied to study the escape…
For a graph $G$ and $p\in [0,1]$, let $G_p$ arise from $G$ by deleting every edge mutually independently with probability $1-p$. The random graph model $(K_n)_p$ is certainly the most investigated random graph model and also known as the…
The existence of Hamiltonian cycles in 1-planar graphs with higher connectivity has attracted considerable attention. Recently, the authors and Dong proved that 4-connected 1-planar chordal graphs are Hamiltonian-connected. In this paper,…
We prove two dichotomy results for detecting long paths as patterns in a given graph. The NP-hard problem Longest Induced Path is to determine the longest induced path in a graph. The NP-hard problem Longest Path Contractibility is to…
We compute the number of circuits and of loops with multiple crossings in random regular graphs. We discuss the importance of this issue for the validity of the cavity approach. On the one side we obtain analytic results for the infinite…