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On $(1,2)$-step competition graphs of multipartite tournaments

Myungho Choi, Suh-Ryung Kim

Abstract

A multipartite tournament is an orientation of a complete $k$-partite graph for some positive integer $k\geq 3$. We say that a multipartite tournament $D$ is tight if every partite set forms a clique in the $(1,2)$-step competition graph, denoted by $C_{1,2}(D)$, of $D$. In this paper, we completely characterize $C_{1,2}(D)$ for a tight multipartite tournament $D$. We will study $C_{1,2}(D)$ for a multipartite tournament $D$ that is not tight in a follow up paper.

On $(1,2)$-step competition graphs of multipartite tournaments

Abstract

A multipartite tournament is an orientation of a complete -partite graph for some positive integer . We say that a multipartite tournament is tight if every partite set forms a clique in the -step competition graph, denoted by , of . In this paper, we completely characterize for a tight multipartite tournament . We will study for a multipartite tournament that is not tight in a follow up paper.
Paper Structure (6 sections, 16 theorems, 29 equations, 1 figure)

This paper contains 6 sections, 16 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $C_{1,2}(D)$ for a tight multipartite tournament $D$ with a maximum anti-$\{1,2\}$-competing set of size one
  4. $C_{1,2}(D)$ for a tight multipartite tournament $D$ with a maximum anti-$\{1,2\}$-competing set of size two
  5. $C_{1,2}(D)$ for a tight multipartite tournament $D$ with a maximum anti-$\{1,2\}$-competing set of size three
  6. Acknowledgement

Key Result

Proposition 2.1

Let $D$ be a multipartite tournament, and $u$ and $v$ be two non-sink vertices belonging to the same partite set in $D$. Then the following are true:

Figures (1)

  • Figure 1: $D_1$, $D_2$, $D_3$, and $D_4$ are orientations of $K_{3,3,1},K_{2,2,2}, K_{3,1,1,1}$, and $K_{2,2,1,1}$, respectively, whose $(1,2)$-step competition graphs are complete

Theorems & Definitions (29)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • ...and 19 more