Related papers: Large cycles in generalized Johnson graphs

We study the variable-length ensemble of self-avoiding walks on the complete graph. We obtain the leading order asymptotics of the mean and variance of the walk length, as the number of vertices goes to infinity. Central limit theorems for…

We give a simple geometric interpretation of an algebraic construction of Wenger that yields $n$-vertex graphs with no cycle of length $4$, $6$ or $10$ and close to the maximum number of edges.

We revisit the problem of counting the number of copies of a fixed graph in a random graph or multigraph, for various models of random (multi)graphs. For our proofs we introduce the notion of \emph{patchworks} to describe the possible…

Let $\Gamma(n,k)$ be the set of $2$-connected $n$-vertex graphs containing an edge that is not on any cycle of length at least $k+1.$ Let $g_s(n,k)$ denote the maximum number of $s$-cliques in a graph in $\Gamma(n,k).$ Recently, Ji and Ye…

We determine the exact and asymptotic number of unlabeled outerplanar graphs. The exact number g_n of unlabeled outerplanar graphs on n vertices can be computed in polynomial time, and g_n is asymptotically $g n^{-5/2}\rho^{-n}$, where…

We initiate the study of the cycle structure of uniformly random parking functions. Using the combinatorics of parking completions, we compute the asymptotic expected value of the number of cycles of any fixed length. We obtain an upper…

General results on asymptotic expansions of Feynman diagrams in momenta and/or masses are reviewed. It is shown how they are applied for calculation of massive diagrams.

There is a sizable literature on investigating the minimum and maximum numbers of cycles in a class of graphs. However, the answer is known only for special classes. This paper presents a result on the smallest number of cycles in…

We give an upper bound on the number of cycles in a simple graph in terms of its degree sequence, and apply this bound to resolve several conjectures of Kir\'aly and Arman and Tsaturian and to improve upper bounds on the maximum number of…

The enumeration of Hamiltonian cycles on 2n*2n grids of nodes is a longstanding problem in combinatorics. Previous work has concentrated on counting all cycles. The current work enumerates nonisomorphic cycles -- that is, the number of…

It is conjectured that every fullerene graph is hamiltonian. Jendrol' and Owens proved [J. Math. Chem. 18 (1995), pp. 83--90] that every fullerene graph on n vertices has a cycle of length at least 4n/5. In this paper, we improve this bound…

Asymptotic expansions for a wide class of distribution are studied. A simple method for computation of the series coefficients is suggested. The case when regularization parameter of the distribution depends on the asymptotic parameter is…

We show that a sequence of graphs with geometric property (T) has many small cycles. We also show that when a small part of a sequence of graphs with geometric property (T) is changed, it still has geometric property (T), provided that it…

We present analytical results for the distribution of the number of cycles in directed and undirected random 2-regular graphs (2-RRGs) consisting of $N$ nodes. In directed 2-RRGs each node has one inbound link and one outbound link, while…

In this paper, we present the lower bounds for the number of vertices in a graph with a large chromatic number containing no small odd cycles.

The objects of our interest are the so-called $A$-permutations, which are permutations whose cycle length lie in a fixed set $A$. They have been extensively studied with respect to the uniform or the Ewens measure. In this paper, we extend…

Generalized cycles can be thought of as the extension of form-cycle duality between holomorphic forms and cycles, to meromorphic forms and generalized cycles. They appeared as an ubiquitous tool in the study of spectral curves and…

We discuss the length of the longest cycle in a sparse random graph $G_{n,p},p=c/n$. $c$ constant. We show that for large $c$ there is a function $f(c)$ such that $L_n(c)/n\to f(c)$ a.s. The function $f(c)=1-\sum_{k=1}^\infty p_k(c)e^{-kc}$…

We obtain bivariate asymptotics for the number of (unicellular) combinatorial maps (a model of discrete surfaces) as both the size and the genus grow. This work is related to two research topics that have been very active recently:…

A number of new sufficient conditions for generalized cycles (large cycles including Hamilton and dominating cycles as special cases) in an arbitrary $k$-connected graph $(k=1,2,...)$ and new lower bounds for the circumference (the length…