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A new class of bi-transversal matroids

Mahdi Ebrahimi

Abstract

A transversal matroid whose dual is also transversal is called bi-transversal. Let $G$ be an undirected graph with vertex set $V$. In this paper, for every subset $W$ of $V$, we associate a bi-transversal matroid to the pair $(G,W)$. We also derive an explicit formula for counting bases of this matroid.

A new class of bi-transversal matroids

Abstract

A transversal matroid whose dual is also transversal is called bi-transversal. Let be an undirected graph with vertex set . In this paper, for every subset of , we associate a bi-transversal matroid to the pair . We also derive an explicit formula for counting bases of this matroid.
Paper Structure (2 sections, 3 theorems, 4 equations, 1 figure)

This paper contains 2 sections, 3 theorems, 4 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Graphical transversal matroids

Key Result

Theorem 1.1

Suppose $G=(V,E)$ is an undirected graph and $S:=\{(v,i)\in V\times \mathbb{N}|\, 1\leqslant i\leqslant \mathrm{deg}_G(v)\}$. Assume that $r$ is a function from $2^{S}$ into $\mathbb{Z}$ defined by $r(X):=max \{\mathrm{Hei}(\varphi)|\,\varphi\,\text{is a labeling of } G \text{ with respect to } \alp

Figures (1)

  • Figure 1: labeling of graphs.

Theorems & Definitions (6)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Example 2.2