Abstract
In this paper, we investigate partitions of highly symmetrical discrete structures called cycloids. In general, a mixed hypergraph has two types of hyperedges. The vertices are colored in such a way that each C-edge has two vertices of the same color, and each D-edge has two vertices of distinct colors. In our case, a mixed cycloid is a mixed hypergraph whose vertices can be arranged in a cyclic order, and every consecutive p vertices form a C-edge, and every consecutive q vertices form a D-edge in the ordering. We completely determine the maximum number of colors that can be used for any p≥3 and any q≥2. We also develop an algorithm that generates a coloring with any number of colors between the minimum and maximum. Finally, we discuss the colorings of mixed cycloids when the maximum number of colors coincides with its upper bound, which is the largest cardinality of a set of vertices containing no C-edge.
Highlights
Academic Editor: Magdalena LemańskaReceived: 23 July 2021Accepted: 17 August 2021Published: 21 August 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.Many processes in different areas of science, technology, engineering, etc., have a circular nature and are modeled by circular structures using graphs and hypergraphs
We considered colorings of complete circular hypergraphs in which the
The main result is the complete determination of the largest possible number of colors under these conditions for any number n of vertices, any p ≥ 3, and any q ≥ 2
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Many processes in different areas of science, technology, engineering, etc., have a circular nature and are modeled by circular structures using graphs and hypergraphs. They are important tools in finding optimal solutions to optimization problems (see [1] for a fresh random example). The general idea is that elements of the structure (called vertices) are placed in cyclic ordering, and intervals of elements in the ordering have a specified cardinality, and all are present. In this model, elements of a structure can be in a number of discrete states called colors. The partitions and restrictions on them are studied using the language of mixed hypergraph coloring, termed as colorings, of complete ( p, q)-uniform circular mixed hypergraphs of order n
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