Research On The Competition Index Of Multipartite Tournaments

Competition graphs are very important graphs in graph theory,which are widely used in the research of communications under noisy channels,radio broadcasting research,and ecosystems.In the process of researching competition graphs,Cho and Kim et al.proposed the concept of competitive index of directed graphs.In recent years,Researchers are interested in the competition index.Tournaments are one of the graphs with the highest research attention and richest content in the directed graph theory.Bang-Jensen et al.described the structure,classification and related conclusions of tournaments.Since then,this type of graph has aroused wide interest among researchers and has obtained many meaningful results.Multipartite tournaments are an important promotion of tournaments.This article mainly studies regular multipartite tournaments,almost regular multipartite tournaments,and the competition index of an extended tournament.By observing whether any two vertices of different part sets and the same part set are adjacent in the m-step competition graph,whether there is a sink,etc.,find the law of the competition index of the graph class under study,and give a proof.This article is divided into four chapters.The first chapter introduces the research content and significance of the competition index of multipartite tournaments,as well as the basic concepts,content arrangements and main results related to this article.The second chapter studies the competition index of regular multipartite tournaments.The third chapter studies the competition index of the almost regular two-part tournaments,the necessary and sufficient conditions when the competition index of the almost regular two-part tournament is 1,and the necessary and sufficient conditions when the competition index of the regular two-part tournament is 1 and 2.The fourth chapter studies the competition index of the extended tournament.The regular multipartite tournament is an important sub-graph category of the multipartite tournament.In the second chapter,by studying the competition index of the regularn(n? 4)-part tournaments,the regular 3-part tournaments and the regular 2-part tournaments,the regularity is finally completely described.The competition graph of regular multipartite tournaments.The almost regular multipartite tournaments are an important sub-graph category of the multipartite tournaments.In Chapter 3,by discussing whether there is a sink,the competition index of the almost regular two-part tournament is studied and portrayed.Furthermore,the paper also analyzes the necessary and sufficient conditions when the competition index of the almost regular two-part tournament is 1 and the necessary and sufficient conditions when the competition index of the regular two-part tournament is 1 and 2.The extended tournament is a relatively extensively researched subgraph category of multipartite tournaments.Through the structural analysis and research of this type of graph,the description of the competition index of the extended tournament is completed.