Modifikace magických ohodnocení

Abstract

Competitiveness is one of the human traits that stays with us throughout our lives. Fairness should be the natural foundation of competitiveness. This thesis focuses on distance magic graph labelings, which are used to construct incomplete sports tournaments. In order to evaluate graphs that encounter algebraic constraints of distance magic labelings, we must generalize the definition. The aim of this thesis is to explore generalizations of the magic labeling that allow us to assign magic weights to the aforementioned graphs with algebraic constraints. In this thesis, we define this generalization as an extended distance magic labeling, which we subsequently use to construct fair incomplete tournaments. We define necessary and sufficient conditions for these evaluations, we construct distance magic labelings using Kotzig arrays and lifted Kotzig arrays, and for the extended distance magic labeling, we introduce modified Kotzig arrays. To ensure that a single team does not play two matches at the same time, we must construct a valid match scheduling using 1-factorization. The thesis demonstrates that, based on the chosen generalization, extended distance magic labeling can be found for previously problematic graphs. In conclusion, we implement these findings and obtain a tool for generating graphs with extended distance magic labelings.