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On The Number Of Labeled Bipartite Graphs

Abdullah Atmaca, A. Yavuz Oruc

Abstract

Let $I$ and $O$ denote two sets of vertices, where $I\cap O =\emptyset$, $|I| = n$, $|O| = r$, and $B_u(n,r)$ denote the set of unlabeled graphs whose edges connect vertices in $I$ and $O$. Recently, it was established in\cite{atmacaoruc2018} that the following two-sided equality holds, \begin{equation} \label{mainResult} \displaystyle \frac{\binom{r+2^{n}-1}{r}}{n!}\, \le\, |B_u(n,r)|\, \le\, 2\frac{\binom{r+2^{n}-1}{r}}{n!},\, n < r.\nonumber \end{equation} and exact formulas were provided in~\cite{atmaca2017size} for $|B_u(2,r)|$ and $|B_u(3,r)|.$ In this paper, these results are extended to various families of labeled bipartite graphs.

On The Number Of Labeled Bipartite Graphs

Abstract

Let and denote two sets of vertices, where , , , and denote the set of unlabeled graphs whose edges connect vertices in and . Recently, it was established in\cite{atmacaoruc2018} that the following two-sided equality holds, \begin{equation} \label{mainResult} \displaystyle \frac{\binom{r+2^{n}-1}{r}}{n!}\, \le\, |B_u(n,r)|\, \le\, 2\frac{\binom{r+2^{n}-1}{r}}{n!},\, n < r.\nonumber \end{equation} and exact formulas were provided in~\cite{atmaca2017size} for and In this paper, these results are extended to various families of labeled bipartite graphs.
Paper Structure (6 sections, 7 theorems, 47 equations, 6 figures)

This paper contains 6 sections, 7 theorems, 47 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Left-Set-Labeled Bipartite Graphs
  4. Set-Labeled Bipartite Graphs
  5. Lower And Upper Bounds for $|B_x(n,r)|$
  6. Lower and Upper Bounds For $|B_{x,y}(n,r)|$

Key Result

Theorem 1

$\,$ where $A(r) = \binom{r+7}{r} + \frac{3\left ( r+4 \right )\left ( 2r^{4}+32r^{3}+172r^{2} + 352r + 15\left ( -1 \right )^{r} +225 \right )}{960},$ and

Figures (6)

  • Figure 1: Examples of equivalent and non-equivalent left-set-labeled $(4,4)$-bipartite graphs.
  • Figure 2: Non-equivalent left-set-labeled (2,1) and (2,2)-bipartite graphs.
  • Figure 3: Non-equivalent left-set-labeled (2,3)-bipartite graphs.
  • Figure 4: Non-equivalent left set-labeled (3,2)-bipartite graphs.
  • Figure 5: Non-equivalent set-labeled (1,2) and (2,2)-bipartite graphs.
  • ...and 1 more figures

Theorems & Definitions (17)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Proof
  • Corollary 1
  • Proof
  • Theorem 3
  • Proof
  • ...and 7 more