scholarly journals Matching Polytons

10.37236/6241 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Martin Doležal ◽  
Jan Hladký
Keyword(s):  
Combinatorial Optimization ◽  
Chromatic Number ◽  
Vertex Cover ◽  
Independent Interest ◽  
Finite Graphs ◽  
Symmetric Spectrum ◽  
Matching Polytope ◽  
Convergent Sequences

Hladký, Hu, and Piguet [Tilings in graphons, preprint] introduced the notions of matching and fractional vertex covers in graphons. These are counterparts to the corresponding notions in finite graphs.  Combinatorial optimization studies the structure of the matching polytope and the fractional vertex cover polytope of a graph. Here, in analogy, we initiate the study of the structure of the set of all matchings and of all fractional vertex covers in a graphon. We call these sets the matching polyton and the fractional vertex cover polyton. We also study properties of matching polytons and fractional vertex cover polytons along convergent sequences of graphons.  As an auxiliary tool of independent interest, we prove that a graphon is $r$-partite if and only if it contains no graph of chromatic number $r+1$. This in turn gives a characterization of bipartite graphons as those having a symmetric spectrum.

A Characterization of Graphs with Fractional Total Chromatic Number Equal to

2009 ◽  
Vol 35 ◽  
pp. 235-240 ◽  
Author(s):  
Takehiro Ito ◽  
W. Sean Kennedy ◽  
Bruce A. Reed
Keyword(s):  
Chromatic Number ◽  
Total Chromatic Number

scholarly journals Two examples related to conical energies

2022 ◽  
Vol 47 (1) ◽  
pp. 261-281
Author(s):  
Damian Dąbrowski
Keyword(s):  
Singular Integral ◽  
Recent Article ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
The Other ◽  
Independent Interest ◽  
Sufficient Condition ◽  
Lipschitz Graphs ◽  
Big Pieces

In a recent article (2021) we introduced and studied conical energies. We used them to prove three results: a characterization of rectifiable measures, a characterization of sets with big pieces of Lipschitz graphs, and a sufficient condition for boundedness of nice singular integral operators. In this note we give two examples related to sharpness of these results. One of them is due to Joyce and Mörters (2000), the other is new and could be of independent interest as an example of a relatively ugly set containing big pieces of Lipschitz graphs.

scholarly journals Commuting decomposition of Kn1,n2,...,nk through realization of the product A(G)A(GPk )

2018 ◽  
Vol 6 (1) ◽  
pp. 343-356
Author(s):  
K. Arathi Bhat ◽  
G. Sudhakara
Keyword(s):  
Chromatic Number ◽  
Perfect Matching ◽  
Domination Number ◽  
Perfect Matchings ◽  
Factor Graph ◽  
Vertex Set ◽  
Graph Parameters

Abstract In this paper, we introduce the notion of perfect matching property for a k-partition of vertex set of given graph. We consider nontrivial graphs G and GPk , the k-complement of graph G with respect to a kpartition of V(G), to prove that A(G)A(GPk ) is realizable as a graph if and only if P satis_es perfect matching property. For A(G)A(GPk ) = A(Γ) for some graph Γ, we obtain graph parameters such as chromatic number, domination number etc., for those graphs and characterization of P is given for which GPk and Γ are isomorphic. Given a 1-factor graph G with 2n vertices, we propose a partition P for which GPk is a graph of rank r and A(G)A(GPk ) is graphical, where n ≤ r ≤ 2n. Motivated by the result of characterizing decomposable Kn,n into commuting perfect matchings [2], we characterize complete k-partite graph Kn1,n2,...,nk which has a commuting decomposition into a perfect matching and its k-complement.

scholarly journals Improved queue-size scaling for input-queued switches via graph factorization

2020 ◽  
Vol 52 (3) ◽  
pp. 798-824
Author(s):  
Jiaming Xu ◽  
Yuan Zhong
Keyword(s):  
Independent Interest ◽  
Queue Size ◽  
Scheduling Policies ◽  
New Class ◽  
Size Scaling ◽  
Size Scales ◽  
Graph Factorization

AbstractThis paper studies the scaling of the expected total queue size in an $n\times n$ input-queued switch, as a function of both the load $\rho$ and the system scale n. We provide a new class of scheduling policies under which the expected total queue size scales as $O\big( n(1-\rho)^{-4/3} \log \big(\!\max\big\{\frac{1}{1-\rho}, n\big\}\big)\big)$, over all n and $\rho<1$, when the arrival rates are uniform. This improves on the best previously known scalings in two regimes: $O\big(n^{1.5}(1-\rho)^{-1} \log \frac{1}{1-\rho}\big)$ when $\Omega\big(n^{-1.5}\big) \le 1-\rho \le O\big(n^{-1}\big)$ and $O\big(\frac{n\log n}{(1-\rho)^2}\big)$ when $1-\rho \geq \Omega(n^{-1})$. A key ingredient in our method is a tight characterization of the largest k-factor of a random bipartite multigraph, which may be of independent interest.

A Combinatorial Optimization Problem for High Order PODs with Few Sensors

2006 ◽  
Vol 129 (2) ◽  
pp. 252-255
Author(s):  
Atanu K. Mohanty ◽  
Kanad Chakraborty ◽  
Anindya Chatterjee
Keyword(s):  
Combinatorial Optimization ◽  
Dynamic Systems ◽  
Optimization Problem ◽  
Combinatorial Optimization Problem ◽  
High Dimensional ◽  
Experimental Characterization ◽  
State Behavior ◽  
Simultaneous Measurements ◽  
Proper Orthogonal

Experimental characterization of high dimensional dynamic systems sometimes uses the proper orthogonal decomposition (POD). If there are many measurement locations and relatively fewer sensors, then steady-state behavior can still be studied by sequentially taking several sets of simultaneous measurements. The number required of such sets of measurements can be minimized if we solve a combinatorial optimization problem. We aim to bring this problem to the attention of engineering audiences, summarize some known mathematical results about this problem, and present a heuristic (suboptimal) calculation that gives reasonable, if not stellar, results.

SPARSE PARTITION REGULARITY

2006 ◽  
Vol 93 (3) ◽  
pp. 545-569 ◽  
Author(s):  
IMRE LEADER ◽  
PAUL A. RUSSELL
Keyword(s):  
Chromatic Number ◽  
Independent Interest ◽  
Natural Numbers ◽  
Ramsey Theorem

Our aim in this paper is to prove Deuber's conjecture on sparse partition regularity, that for every $m$, $p$ and $c$ there exists a subset of the natural numbers whose $(m,p,c)$-sets have high girth and chromatic number. More precisely, we show that for any $m$, $p$, $c$, $k$ and $g$ there is a subset $S$ of the natural numbers that is sufficiently rich in $(m,p,c)$-sets that whenever $S$ is $k$-coloured there is a monochromatic $(m,p,c)$-set, yet is so sparse that its $(m,p,c)$-sets do not form any cycles of length less than $g$.Our main tools are some extensions of Nešetřil–Rödl amalgamation and a Ramsey theorem of Bergelson, Hindman and Leader. As a sideline, we obtain a Ramsey theorem for products of trees that may be of independent interest.

scholarly journals Dunford-Pettis and strongly Dunford-Pettis operators on L1(μ)

1989 ◽  
Vol 31 (1) ◽  
pp. 49-57 ◽  
Author(s):  
James R. Holub
Keyword(s):  
Positive Operator ◽  
Finite Measure ◽  
Regular Operator ◽  
Mathematical Economics ◽  
Canonical Injection ◽  
Do So ◽  
Convergent Sequences

Motivated by a problem in mathematical economics [4] Gretsky and Ostroy have shown [5] that every positive operator T:L1[0, 1] → c0 is a Dunford-Pettis operator (i.e. T maps weakly convergent sequences to norm convergent ones), and hence that the same is true for every regular operator from L1[0, 1] to c0. In a recent paper [6] we showed the converse also holds, thereby characterizing the D–P operators by this condition. In each case the proof depends (as do so many concerning D–P operators on Ll[0, 1]) on the following well-known result (see, e.g., [2]): If μ is a finite measure, an operator T:L1(μ) → E is a D–P operator is compact, where i:L∞(μ) → L1(μ) is the canonical injection of L∞(μ) into L1(μ). If μ is not a finite measure this characterization of D–P operators is no longer available, and hence results based on its use (e.g. [5], [6]) do not always have straightforward extensions to the case of operators on more general L1(μ) spaces.

scholarly journals A constructive characterization of vertex cover Roman trees

2021 ◽  
Vol 41 (1) ◽  
pp. 267 ◽  
Author(s):  
Abel Cabrera Martínez ◽  
Dorota Kuziak ◽  
González Ismael Yero
Keyword(s):  
Vertex Cover
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