## Control Systems and Computers, N5, 2018, Article 2

##### DOI: https://doi.org/10.15407/usim.2018.05.013

Upr. sist. maš., 2018, Issue 5 (277), pp. 13-24.

UDC 519.1

Semeniuta Marina F., PhD in Phys.-Math. Sciences, professor Department of Physics and Mathematics Sciences of the Flight Academy of the National Aviation University, st. Dobrovolsky, 1, Kropivnitsky, 25005, Ukraine, marina_semenyuta@ukr.net

Sherman Zoya A., PhD in Phys.-Math. Sciences, senior lecturer, Department of Medical physics and information technology №2 of Donetsk National Medical University, st. Pryvokzalnaya, 27, Liman, 84404, Donetsk region, Ukraine,
E-mail: sherman.zoya@gmail.com

Dmitriiev Oleh M., PhD, head of the department of flight operations, aerodynamics and flight dynamics of the Flight Academy of the National Aviation University, E-mail – Dmitronik70@i.ua

### Incomplete tournaments and magic types of labeling

Introduction. There is a variety of types of sports tournaments whose planning is based on the construction of graph models. Each type of tournament has its own characteristics. In this paper, fair, equitable and balanced partial competitions are considered. Creating a tournament grid for n teams playing with r opponents for such tournaments is equivalent to solving the problem of constructing an appropriate distant magic or antimagic labeling of the r-regular graph of order n.

Purpose. The purpose of the article is to systematize the main theoretical information related to this topic, to highlight the problems that have not been solved, to classify the methods of constructing graphs of tournaments and to unify the algorithms for their description in accordance with this classification.

Methods. New algorithms for constructing incomplete tournaments graphs are offered. This makes it possible to extend the range of tasks using mathematical models based on labeled graphs.

Results. All methods of constructing graphs of tournaments are divided into three groups. The methods of the first group included those based on the properties of magic rectangles, including Kotsih arrays. Methods of the second and third groups are constructive and contain elements of induction. Each group is related to the definition of a particular factor or factorization of the graph, which is involved in building a graph of the tournament.

Conclusion. In the process of analyzing the theoretical advances of the studied problem the systematization of the existing results has been made. All methods for constructing incomplete tournament graphs are divided into three groups. New approaches for their realization are offered. This makes it possible to extend the range of tasks using mathematical models based on labeled graphs.

Keywords: fair incomplete tournament, equalized incomplete tournament, handicap tournament, distance magic labeling, d-antimagic distance labeling, d-handicap distance antimagic labeling.

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