スペクトラルグラフ理論を用いた離散構造とアルゴリズムの研究

科学研究費助成事業 研究成果報告書：若手研究(B)2015-2017課題番号 : 15K2088

The triangle-free graphs which are competition graphs of multipartite tournaments

In this paper, we show that a connected triangle-free graph is the competition graph of a $k$-partite tournament if and only if $k \in \{3,4,5\}$, and a disconnected triangle-free graph is the competition graph of a $k$-partite tournament if and only if $k \in \{2,3,4\}$. Then we list all the triangle-free graphs in each case.Comment: 26 pages, 12 figure

Multipartite tournaments whose competition graphs are complete

In this paper, we completely characterize the multipartite tournaments whose competition graphs are complete

방향 지어진 완전 이분 그래프의 m-step 경쟁 그래프

학위논문 (석사)-- 서울대학교 대학원 : 사범대학 수학교육과, 2018. 2. 김서령.In this thesis, we study the m-step competition graphs of bipartite tournaments. We compute the number of distinct bipartite tournaments by Polya's theory of counting. Then we study the competition indices and competition periods of bipartite tournaments. We characterize the pairs of graphs that can be represented as the m-step competition graphs of bipartite tournaments. Finally, we present the maximum number of edges and the minimum number of edges which the m-step competition graph of a bipartite tournament might have.1 Introduction 1 1.1 Basic graph terminology 1 1.2 Competition graph and its variants 3 1.3 m-step competition graphs 5 1.4 Polya's theory of counting 6 1.5 Competition indices and competition periods 9 1.6 Preview of thesis 10 2 The number of distinct bipartite tournaments 12 3 Properties of m-step competition graphs of bipartite tournaments 18 4 m-step competition realizable pairs 23 5 Extremal cases 31 6 Concluding remarks 36Maste

방향 지어진 완전이분그래프의 (i,j)-step 경쟁그래프

학위논문 (석사)-- 서울대학교 대학원 : 수학교육과, 2015. 8. 김서령.In this thesis, we study the (i,j)-step competition graph of an oriented complete bipartite graph. Kim et al. studied the competition graph of an oriented complete bipartite graph. We take a further step to study the (1,2)-step competition graph of an oriented complete bipartite graph by extending the results given by Kim et al. Then we study (i,j)-step competition graph, a more general version of (1,2)-step competition graph. Finally, we deal with the limit of (i,j)-step competition graph.Abstract 1 Introduction 1.1 The notion of (1,2)-step competition graphs 1.2 The competition graphs of oriented complete bipartite graphs 1.3 A preview of thesis 1.4 (i,j)-step competition graphs 2 (1,2)-step competition graphs 2.1 A characterization of (1,2)-step competition graphs 2.2 Structural characterization of (1,2)-step competition graphs 2.3 Extremal (1,2)-step competition graphs 3 (i,j)-step competition graphs 4 Closing remarks Abstract(in Korean)Maste

On niche graphs of bipartite tournaments

학위논문 (석사)-- 서울대학교 대학원 : 사범대학 수학교육과, 2018. 2. 김서령.Let D be a digraph. The niche graph of D has the same set of vertices as D and an edge between vertices u and v if and only if there exists a common in-neighbor or a ommon out-neighbor of u and v in D. Kim et al. [The competition graphs of oriented complete bipartite graphs, Discrete Applied Mathematics 201 (2016) 182–190] studied the competition graphs of bipartite tournaments. In this thesis, we study the niche graphs of bipartite tournaments to extend their results. We characterize graphs that can be represented as the niche graphs of bipartite tournaments. Then we present forbidden induced subgraphs for niche graphs of bipartite tournaments. We also study niche graphs of strongly connected bipartite tournaments. Finally, we consider the extremal cases of niche graphs of bipartite tournaments each of which has the maximum number of edges or the minimum number of edges.1 Introduction 1 1.1 Basic notions in graph theory 1 1.2 Competition graphs and its variants 3 1.3 A preview of thesis 5 2 Niche graphs of bipartite tournamnets 6 2.1 Fundamental structures of niche graphs of bipartite tournamnets 6 2.2 Forbidden induced subgraphs for niche graphs of bipartite tournaments 9 2.3 Niche-realizable pairs 12 3 Niche graphs of strongly connected bipartite tournaments 21 3.1 Strongly niche-realizable pairs 21 3.2 A relationship between directed cycles in a bipartite tournament D with bipartition (U, V) and the connectedness of N(D)[U] 27 4 Extremal cases 29 5 Concluding remarks 33 Abstract (in Korean) 38Maste