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A survey of bipartite tournaments | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 26 تیر 1404 | ||
| نوع مقاله: Survey paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2025.30326.2415 | ||
| نویسندگان | ||
| Jay S. Bagga* 1؛ Lowell W. Beineke2 | ||
| 1Ball State University, Muncie, Indiana, USA | ||
| 2Purdue University Fort Wayne, Fort Wayne, Indiana, USA | ||
| چکیده | ||
| The family of mathematical objects known as tournaments constitutes one of the main areas of directed graphs, being defined as having a set of vertices in which each pair is joined by exactly one arc. Here we are interested in a secondary family known as bipartite tournaments. These are defined as having two non-empty sets of vertices and one arc joining each pair that are in different sets. There is a substantial theory of this topic also, and this article presents many of its foundational areas. One key concept is that of the number of arcs going from a vertex, called its score, and the theory involves what lists of numbers can constitute the scores. The cycles in a bipartite tournament have a variety of interesting collections involving lengths. Other fundamental concepts that we discuss include bipartite tournaments with the property of being self-converse, the efficient removal of cycles by the reversal of arcs, development of a theory of upsets, and properties of automorphism groups of the structures. The concept of bipartite tournaments also generalizes naturally to multipartite tournaments with more parts. Many of the results we discuss in this paper lead to questions about analogous extensions to these structures. In the last section of the paper, we include a discussion of further research directions. | ||
| کلیدواژهها | ||
| Directed graph؛ Tournament؛ Bipartite tournament | ||
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