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An Asymptotically Sharp Bound on the Maximum Number of Independent Transversals

Jake Ruotolo, Kevin Wang, Fan Wei

Abstract

Let $G$ be a multipartite graph with partition $V_1, V_2,\ldots, V_k$ of $V(G)$. Let $d_{i,j}$ denote the edge density of the pair $(V_i, V_j)$. An independent transversal is an independent set of $G$ with exactly one vertex in each $V_i$. In this paper, we prove an asymptotically sharp upper bound on the maximum number of independent transversals given the $d_{i,j}$'s.

An Asymptotically Sharp Bound on the Maximum Number of Independent Transversals

Abstract

Let be a multipartite graph with partition of . Let denote the edge density of the pair . An independent transversal is an independent set of with exactly one vertex in each . In this paper, we prove an asymptotically sharp upper bound on the maximum number of independent transversals given the 's.
Paper Structure (4 sections, 8 theorems, 29 equations, 1 figure)

This paper contains 4 sections, 8 theorems, 29 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm:main']}
  3. Proof of Upper Bound
  4. Proof of Asymptotic Sharpness

Key Result

Theorem 1.1

Let $k \geq 2$ be an integer. For each integer pair $1\leq i < j \leq k$, let $d_{i,j} = d_{j,i}$ be constants in $[0,1]$. Let $G$ be a multipartite graph with vertex partition $V_1, V_2, \ldots, V_k$ such that for each pair $1 \leq i < j \leq k$, the edge density between $V_i, V_j$ is $d_{i,j}$. Le Then the number of independent transversals in $G$ is at most

Figures (1)

  • Figure 1: An odd cycle decomposition of $K_8$.

Theorems & Definitions (42)

  • Theorem 1.1: Lemma 4.1 FHL
  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Claim 2.2
  • proof
  • Claim 2.3
  • proof
  • Claim 2.4
  • ...and 32 more