scholarly journals On Star Forest Ascending Subgraph Decomposition

10.37236/5380 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Josep M. Aroca ◽  
Anna Lladó
Keyword(s):  
Bipartite Graph ◽  
Degree Sequence ◽  
Necessary Condition ◽  
Stable Sets ◽  
Ascending Subgraph Decomposition ◽  
Edge Decomposition

The Ascending Subgraph Decomposition (ASD) Conjecture asserts that every graph $G$ with ${n+1\choose 2}$ edges admits an edge decomposition $G=H_1\oplus\cdots \oplus H_n$ such that $H_i$ has $i$ edges and it is isomorphic to a subgraph of $H_{i+1}$, $i=1,\ldots ,n-1$. We show that every bipartite graph $G$ with ${n+1\choose 2}$ edges such that the degree sequence $d_1,\ldots ,d_k$ of one of the stable sets satisfies $ d_{k-i}\ge n-i\; \text{for each}\; 0\le i\le k-1$, admits an ascending subgraph decomposition with star forests. We also give a necessary condition on the degree sequence which is not far from the above sufficient one.

scholarly journals Greedoids on vertex sets of B-joins of graphs

2014 ◽  
Vol 30 (3) ◽  
pp. 335-344
Author(s):  
VADIM E. LEVIT ◽  
◽  
EUGEN MANDRESCU ◽  
Keyword(s):  
Bipartite Graph ◽  
Sufficient Conditions ◽  
Local Maximum ◽  
Stable Set ◽  
Stable Sets ◽  
The Family ◽  
Maximum Stable Set ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
Vertex Sets

Let Ψ(G) be the family of all local maximum stable sets of graph G, i.e., S ∈ Ψ(G) if S is a maximum stable set of the subgraph induced by S ∪ N(S), where N(S) is the neighborhood of S. It was shown that Ψ(G) is a greedoid for every forest G [15]. The cases of bipartite graphs, triangle-free graphs, and well-covered graphs, were analyzed in [16, 17, 18, 19, 20, 24]. If G1, G2 are two disjoint graphs, and B is a bipartite graph having E(B) as an edge set and bipartition {V (G1), V (G2)}, then by B-join of G1, G2 we mean the graph B (G1, G2) whose vertex set is V (G1) ∪ V (G2) and edge set is E(G1) ∪ E(G2) ∪ E (B). In this paper we present several necessary and sufficient conditions for Ψ(B (G1, G2)) to form a greedoid, an antimatroid, and a matroid, in terms of Ψ(G1), Ψ(G2) and E (B).

scholarly journals Embedding Spanning Bipartite Graphs of Small Bandwidth

2012 ◽  
Vol 22 (1) ◽  
pp. 71-96 ◽  
Author(s):  
FIACHRA KNOX ◽  
ANDREW TREGLOWN
Keyword(s):  
Bipartite Graph ◽  
Minimum Degree ◽  
Degree Sequence ◽  
Bipartite Graphs ◽  
Type Result ◽  
Bounded Degree ◽  
Expansion Property ◽  
Small Bandwidth ◽  
Bipartite Case

Böttcher, Schacht and Taraz (Math. Ann., 2009) gave a condition on the minimum degree of a graph G on n vertices that ensures G contains every r-chromatic graph H on n vertices of bounded degree and of bandwidth o(n), thereby proving a conjecture of Bollobás and Komlós (Combin. Probab. Comput., 1999). We strengthen this result in the case when H is bipartite. Indeed, we give an essentially best-possible condition on the degree sequence of a graph G on n vertices that forces G to contain every bipartite graph H on n vertices of bounded degree and of bandwidth o(n). This also implies an Ore-type result. In fact, we prove a much stronger result where the condition on G is relaxed to a certain robust expansion property. Our result also confirms the bipartite case of a conjecture of Balogh, Kostochka and Treglown concerning the degree sequence of a graph which forces a perfect H-packing.

scholarly journals Edge-Decomposition of Graphs into Copies of a Tree with Four Edges

10.37236/2110 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
János Barát ◽  
Dániel Gerbner
Keyword(s):  
Natural Number ◽  
Connected Graph ◽  
Degree Sequence ◽  
Decomposition Theorem ◽  
Bipartite Graphs ◽  
Connected Graphs ◽  
Edge Decomposition

We study edge-decompositions of highly connected graphs into copies of a given tree. In particular we attack the following conjecture by Barát and Thomassen: for each tree $T$, there exists a natural number $k_T$ such that if $G$ is a $k_T$-edge-connected graph, and $|E(T)|$ divides $|E(G)|$, then $E(G)$ has a decomposition into copies of $T$. As one of our main results it is sufficient to prove the conjecture for bipartite graphs. The same result has been independently obtained by Carsten Thomassen (2013).Let $Y$ be the unique tree with degree sequence $(1,1,1,2,3)$. We prove that if $G$ is a $191$-edge-connected graph of size divisible by $4$, then $G$ has a $Y$-decomposition. This is the first instance of such a theorem, in which the tree is different from a path or a star. Recently Carsten Thomassen proved a more general decomposition theorem for bistars, which yields the same result with a worse constant.

scholarly journals Rejection sampling of bipartite graphs with given degree sequence

2018 ◽  
Vol 10 (2) ◽  
pp. 249-275
Author(s):  
Koko K. Kayibi ◽  
U. Samee ◽  
S. Pirzada ◽  
Mohammad Ali Khan
Keyword(s):  
Markov Chains ◽  
Bipartite Graph ◽  
Degree Sequence ◽  
Bipartite Graphs ◽  
Sampling Scheme ◽  
Rejection Sampling ◽  
Generation Algorithm ◽  
Running Time ◽  
Edge Swapping ◽  
Simple Realization

Abstract Let A = (a1, a2, ..., an) be a degree sequence of a simple bipartite graph. We present an algorithm that takes A as input, and outputs a simple bipartite realization of A, without stalling. The running time of the algorithm is ⊝(n1n2), where ni is the number of vertices in the part i of the bipartite graph. Then we couple the generation algorithm with a rejection sampling scheme to generate a simple realization of A uniformly at random. The best algorithm we know is the implicit one due to Bayati, Kim and Saberi (2010) that has a running time of O(mamax), where $m = {1 \over 2}\sum\nolimits_{i = 1}^n {{a_i}} and amax is the maximum of the degrees, but does not sample uniformly. Similarly, the algorithm presented by Chen et al. (2005) does not sample uniformly, but nearly uniformly. The realization of A output by our algorithm may be a start point for the edge-swapping Markov Chains pioneered by Brualdi (1980) and Kannan et al.(1999).

scholarly journals Symmetric bipartite graphs and graphs with loops

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Grant Cairns ◽  
Stacey Mendan
Keyword(s):  
Graph Theory ◽  
Bipartite Graph ◽  
Degree Sequence ◽  
Bipartite Graphs ◽  
Degree Sequences ◽  
Graph Automorphism ◽  
International Audience ◽  
The Relationship

Graph Theory International audience We show that if the two parts of a finite bipartite graph have the same degree sequence, then there is a bipartite graph, with the same degree sequences, which is symmetric, in that it has an involutive graph automorphism that interchanges its two parts. To prove this, we study the relationship between symmetric bipartite graphs and graphs with loops.

scholarly journals An Entropy Proof of the Kahn-Lovász Theorem

10.37236/497 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Jonathan Cutler ◽  
A. J. Radcliffe
Keyword(s):  
Bipartite Graph ◽  
Upper Bound ◽  
Degree Sequence ◽  
Perfect Matchings ◽  
Entropy Methods ◽  
General Graphs

Brègman gave a best possible upper bound for the number of perfect matchings in a balanced bipartite graph in terms of its degree sequence. Recently Kahn and Lovász extended Brègman's theorem to general graphs. In this paper, we use entropy methods to give a new proof of the Kahn-Lovász theorem. Our methods build on Radhakrishnan's use of entropy to prove Brègman's theorem.

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.

The Infancy of Solar-Type Stars and its Relevance for the Origin of Life

1997 ◽  
Vol 161 ◽  
pp. 267-282 ◽  
Author(s):  
Thierry Montmerle
Keyword(s):  
Main Sequence ◽  
Solar Nebula ◽  
Circumstellar Disks ◽  
T Tauri Stars ◽  
Necessary Condition ◽  
Sequence Evolution ◽  
T Tauri ◽  
Solar Type ◽  
Optical Domain ◽  
The Origin Of Life

AbstractFor life to develop, planets are a necessary condition. Likewise, for planets to form, stars must be surrounded by circumstellar disks, at least some time during their pre-main sequence evolution. Much progress has been made recently in the study of young solar-like stars. In the optical domain, these stars are known as «T Tauri stars». A significant number show IR excess, and other phenomena indirectly suggesting the presence of circumstellar disks. The current wisdom is that there is an evolutionary sequence from protostars to T Tauri stars. This sequence is characterized by the initial presence of disks, with lifetimes ~ 1-10 Myr after the intial collapse of a dense envelope having given birth to a star. While they are present, about 30% of the disks have masses larger than the minimum solar nebula. Their disappearance may correspond to the growth of dust grains, followed by planetesimal and planet formation, but this is not yet demonstrated.

Universal ESEM

1993 ◽  
Vol 51 ◽  
pp. 786-787
Author(s):  
G.D. Danilatos
Keyword(s):  
Electron Microscope ◽  
Scanning Electron Microscope ◽  
High Vacuum ◽  
Natural Extension ◽  
Necessary Condition ◽  
Environmental Scanning ◽  
Detection Systems ◽  
Small Gain ◽  
Detection Medium ◽  
Scanning Electron

The environmental scanning electron microscope (ESEM) has evolved as the natural extension of the scanning electron microscope (SEM), both historically and technologically. ESEM allows the introduction of a gaseous environment in the specimen chamber, whereas SEM operates in vacuum. One of the detection systems in ESEM, namely, the gaseous detection device (GDD) is based on the presence of gas as a detection medium. This might be interpreted as a necessary condition for the ESEM to remain operational and, hence, one might have to change instruments for operation at low or high vacuum. Initially, we may maintain the presence of a conventional secondary electron (E-T) detector in a "stand-by" position to switch on when the vacuum becomes satisfactory for its operation. However, the "rough" or "low vacuum" range of pressure may still be considered as inaccessible by both the GDD and the E-T detector, because the former has presumably very small gain and the latter still breaks down.

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