Monotonicity inequalities for ther-area and a degeneracy theorem forr-minimal graphs

2004 ◽  
Vol 14 (4) ◽  
pp. 557-566 ◽  
Author(s):  
Cleon S. Barroso ◽  
Levi L. de Lima ◽  
Walcy Santos
Keyword(s):  
Minimal Graphs

On Ramsey Minimal Graphs

10.37236/1184 ◽  
1994 ◽  
Vol 1 (1) ◽  
Author(s):  
Tomasz Łuczak
Keyword(s):  
Infinite Number ◽  
Probabilistic Argument ◽  
Minimal Graphs

An elementary probabilistic argument is presented which shows that for every forest $F$ other than a matching, and every graph $G$ containing a cycle, there exists an infinite number of graphs $J$ such that $J\to (F,G)$ but if we delete from $J$ any edge $e$ the graph $J-e$ obtained in this way does not have this property.

On Ramsey $$(mK_2,H)$$ ( m K 2 , H ) -Minimal Graphs

2016 ◽  
Vol 33 (1) ◽  
pp. 233-243 ◽  
Author(s):  
Kristiana Wijaya ◽  
Edy Tri Baskoro ◽  
Hilda Assiyatun ◽  
Djoko Suprijanto
Keyword(s):  
Minimal Graphs

scholarly journals Densities of Minor-Closed Graph Families

10.37236/408 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
David Eppstein
Keyword(s):  
Order Type ◽  
Closed Graph ◽  
Simple Graphs ◽  
Minimal Graphs ◽  
Vertex Graph ◽  
Graph Families ◽  
A Minor ◽  
Cluster Points

We define the limiting density of a minor-closed family of simple graphs $\mathcal{F}$ to be the smallest number $k$ such that every $n$-vertex graph in $\mathcal{F}$ has at most $kn(1+o(1))$ edges, and we investigate the set of numbers that can be limiting densities. This set of numbers is countable, well-ordered, and closed; its order type is at least $\omega^\omega$. It is the closure of the set of densities of density-minimal graphs, graphs for which no minor has a greater ratio of edges to vertices. By analyzing density-minimal graphs of low densities, we find all limiting densities up to the first two cluster points of the set of limiting densities, $1$ and $3/2$. For multigraphs, the only possible limiting densities are the integers and the superparticular ratios $i/(i+1)$.

scholarly journals On Ramsey-Minimal Infinite Graphs

2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Jordan Barrett ◽  
Valentino Vito
Keyword(s):  
Ramsey Theory ◽  
Infinite Graphs ◽  
Minimal Graph ◽  
Finite Graphs ◽  
Minimal Graphs ◽  
Finite Case ◽  
Ramsey Minimal Graph ◽  
General Conditions

For fixed finite graphs $G$, $H$, a common problem in Ramsey theory is to study graphs $F$ such that $F \to (G,H)$, i.e. every red-blue coloring of the edges of $F$ produces either a red $G$ or a blue $H$. We generalize this study to infinite graphs $G$, $H$; in particular, we want to determine if there is a minimal such $F$. This problem has strong connections to the study of self-embeddable graphs: infinite graphs which properly contain a copy of themselves. We prove some compactness results relating this problem to the finite case, then give some general conditions for a pair $(G,H)$ to have a Ramsey-minimal graph. We use these to prove, for example, that if $G=S_\infty$ is an infinite star and $H=nK_2$, $n \geqslant 1$ is a matching, then the pair $(S_\infty,nK_2)$ admits no Ramsey-minimal graphs.

scholarly journals Finite element algorithms for nonlocal minimal graphs

2021 ◽  
Vol 4 (2) ◽  
pp. 1-29
Author(s):  
Juan Pablo Borthagaray ◽  
◽  
Wenbo Li ◽  
Ricardo H. Nochetto ◽  
◽  
...  
Keyword(s):  
Mean Curvature ◽  
Numerical Experiments ◽  
Piecewise Linear ◽  
Gradient Flow ◽  
Minimal Graphs ◽  
Qualitative And Quantitative ◽  
Dirichlet Data ◽  
Data Truncation ◽  
Discrete Problems ◽  
Newton Scheme

<abstract><p>We discuss computational and qualitative aspects of the fractional Plateau and the prescribed fractional mean curvature problems on bounded domains subject to exterior data being a subgraph. We recast these problems in terms of energy minimization, and we discretize the latter with piecewise linear finite elements. For the computation of the discrete solutions, we propose and study a gradient flow and a Newton scheme, and we quantify the effect of Dirichlet data truncation. We also present a wide variety of numerical experiments that illustrate qualitative and quantitative features of fractional minimal graphs and the associated discrete problems.</p></abstract>

scholarly journals On Ramsey (P 3, C 6)-minimal graphs for certain order

2020 ◽  
Vol 1538 ◽  
pp. 012011
Author(s):  
F Nisa ◽  
D Rahmadani ◽  
Purwanto ◽  
H Susanto
Keyword(s):  
Minimal Graphs
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