scholarly journals Asymptotic enumeration of non-uniform linear hypergraphs

2019 ◽  
Author(s):  
Mahdieh Hasheminezhad ◽  
Brendan D. Mckay
Keyword(s):  
Asymptotic Enumeration ◽  
Linear Hypergraphs

scholarly journals Asymptotic Enumeration of Sparse Uniform Linear Hypergraphs with Given Degrees

10.37236/5512 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Vladimir Blinovsky ◽  
Catherine Greenhill
Keyword(s):  
Bipartite Graph ◽  
Maximum Degree ◽  
Incidence Matrix ◽  
Degree Sequence ◽  
Bipartite Graphs ◽  
Asymptotic Enumeration ◽  
Uniform Hypergraphs ◽  
Enumeration Formula ◽  
Linear Hypergraphs

A hypergraph is simple if it has no loops and no repeated edges, and a hypergraph is linear if it is simple and each pair of edges intersects in at most one vertex. For $n\geq 3$, let $r= r(n)\geq 3$ be an integer and let $\boldsymbol{k} = (k_1,\ldots, k_n)$ be a vector of nonnegative integers, where each $k_j = k_j(n)$ may depend on $n$. Let $M = M(n) = \sum_{j=1}^n k_j$ for all $n\geq 3$, and define the set $\mathcal{I} = \{ n\geq 3 \mid r(n) \text{ divides } M(n)\}$. We assume that $\mathcal{I}$ is infinite, and perform asymptotics as $n$ tends to infinity along $\mathcal{I}$. Our main result is an asymptotic enumeration formula for linear $r$-uniform hypergraphs with degree sequence $\boldsymbol{k}$. This formula holds whenever the maximum degree $k_{\max}$ satisfies $r^4 k_{\max}^4(k_{\max} + r) = o(M)$. Our approach is to work with the incidence matrix of a hypergraph, interpreted as the biadjacency matrix of a bipartite graph, enabling us to apply known enumeration results for bipartite graphs. This approach also leads to a new asymptotic enumeration formula for simple uniform hypergraphs with specified degrees, and a result regarding the girth of random bipartite graphs with specified degrees.

scholarly journals Compositions into Powers of b: Asymptotic Enumeration and Parameters

Algorithmica ◽  
2015 ◽  
Vol 75 (4) ◽  
pp. 606-631 ◽  
Author(s):  
Daniel Krenn ◽  
Stephan Wagner
Keyword(s):  
Asymptotic Enumeration

scholarly journals Intersection Graphs of k-uniform Linear Hypergraphs

1982 ◽  
Vol 3 (2) ◽  
pp. 159-172 ◽  
Author(s):  
Ranjan N. Naik ◽  
S.B. Rao ◽  
S.S. Shrikhande ◽  
N.M. Singhi
Keyword(s):  
Intersection Graphs ◽  
Linear Hypergraphs

scholarly journals Asymptotic Enumeration of Tournaments with a Given Score Sequence

1996 ◽  
Vol 73 (1) ◽  
pp. 77-90 ◽  
Author(s):  
Brendan D. McKay ◽  
Xiaoji Wang
Keyword(s):  
Asymptotic Enumeration

scholarly journals A bijection for nonorientable general maps

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Jérémie Bettinelli
Keyword(s):  
Simple Form ◽  
Asymptotic Enumeration ◽  
International Audience ◽  
Enumeration Formula

International audience We give a different presentation of a recent bijection due to Chapuy and Dołe ̨ga for nonorientable bipartite quadrangulations and we extend it to the case of nonorientable general maps. This can be seen as a Bouttier–Di Francesco–Guitter-like generalization of the Cori–Vauquelin–Schaeffer bijection in the context of general nonori- entable surfaces. In the particular case of triangulations, the encoding objects take a particularly simple form and we recover a famous asymptotic enumeration formula found by Gao.

scholarly journals Some Results on Chromaticity of Quasi-Linear Paths and Cycles

10.37236/2370 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Ioan Tomescu
Keyword(s):  
Linear Hypergraphs

Let $r\geq 1$ be an integer. An $h$-hypergraph $H$ is said to be $r$-quasi-linear (linear for $r=1$) if any two edges of $H$ intersect in 0 or $r$ vertices. In this paper it is shown that $r$-quasi-linear paths $P_{m}^{h,r}$ of length $m\geq 1$ and cycles $C_{m}^{h,r}$ of length $m\geq 3$ are chromatically unique in the set of $h$-uniform $r$-quasi-linear hypergraphs provided $r\geq 2$ and $h\geq 3r-1$. 

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