scholarly journals Asymptotic bounds on total domination in regular graphs

2021 ◽  
Vol 344 (4) ◽  
pp. 112287
Author(s):  
Carlos Hoppen ◽  
Giovane Mansan
Keyword(s):  
Regular Graphs ◽  
Total Domination ◽  
Asymptotic Bounds

scholarly journals Isoperimetric Numbers of Regular Graphs of High Degree with Applications to Arithmetic Riemann Surfaces

10.37236/651 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Dominic Lanphier ◽  
Jason Rosenhouse
Keyword(s):  
Large Class ◽  
Lower Bounds ◽  
Riemann Surfaces ◽  
Upper And Lower Bounds ◽  
Regular Graphs ◽  
Eigenvalue Spectrum ◽  
Asymptotic Bounds ◽  
High Degree

We derive upper and lower bounds on the isoperimetric numbers and bisection widths of a large class of regular graphs of high degree. Our methods are combinatorial and do not require a knowledge of the eigenvalue spectrum. We apply these bounds to random regular graphs of high degree and the Platonic graphs over the rings $\mathbb{Z}_n$. In the latter case we show that these graphs are generally non-Ramanujan for composite $n$ and we also give sharp asymptotic bounds for the isoperimetric numbers. We conclude by giving bounds on the Cheeger constants of arithmetic Riemann surfaces. For a large class of these surfaces these bounds are an improvement over the known asymptotic bounds.

scholarly journals The analysis of a prioritised probabilistic algorithm to find large induced forests in regular graphs with large girth

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Carlos Hoppen
Keyword(s):  
Lower Bounds ◽  
Regular Graphs ◽  
Probabilistic Algorithms ◽  
Induced Subgraph ◽  
Fixed Integer ◽  
Asymptotic Bounds ◽  
Random Regular Graph ◽  
International Audience ◽  
Large Girth ◽  
Deterministic Bounds

International audience The analysis of probabilistic algorithms has proved to be very successful for finding asymptotic bounds on parameters of random regular graphs. In this paper, we show that similar ideas may be used to obtain deterministic bounds for one such parameter in the case of regular graphs with large girth. More precisely, we address the problem of finding a large induced forest in a graph $G$, by which we mean an acyclic induced subgraph of $G$ with a lot of vertices. For a fixed integer $r \geq 3$, we obtain new lower bounds on the size of a maximum induced forest in graphs with maximum degree $r$ and large girth. These bounds are derived from the solution of a system of differential equations that arises naturally in the analysis of an iterative probabilistic procedure to generate an induced forest in a graph. Numerical approximations suggest that these bounds improve substantially the best previous bounds. Moreover, they improve previous asymptotic lower bounds on the size of a maximum induced forest in a random regular graph.

scholarly journals Total Domination in Regular Graphs

2019 ◽  
Vol 346 ◽  
pp. 523-533
Author(s):  
Carlos Hoppen ◽  
Giovane Mansan
Keyword(s):  
Regular Graphs ◽  
Total Domination

scholarly journals Upper minus total domination in small-degree regular graphs

2007 ◽  
Vol 307 (21) ◽  
pp. 2453-2463 ◽  
Author(s):  
Hong Yan ◽  
Xiaoqi Yang ◽  
Erfang Shan
Keyword(s):  
Small Degree ◽  
Regular Graphs ◽  
Total Domination

scholarly journals Total domination versus paired-domination in regular graphs

2018 ◽  
Vol 38 (2) ◽  
pp. 573 ◽  
Author(s):  
Joanna Cyman ◽  
Magda Dettlaff ◽  
Michael .A. Henning ◽  
Magdalena Lemańska ◽  
Joanna Raczek
Keyword(s):  
Regular Graphs ◽  
Total Domination ◽  
Paired Domination

Upper Bounds for k-Tuple (Total) Domination Numbers of Regular Graphs

2019 ◽  
Vol 46 (2) ◽  
pp. 573-577
Author(s):  
Sharareh Alipour ◽  
Amir Jafari ◽  
Morteza Saghafian
Keyword(s):  
Upper Bounds ◽  
Regular Graphs ◽  
Total Domination ◽  
Domination Numbers
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