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The minimum size of 2-connected chordal bipartite graphs

Licheng Zhang, Yuanqiu Huang

Abstract

A bipartite graph is chordal bipartite if every cycle of length at least six contains a chord. We determine the minimum size in 2-connected chordal bipartite graphs with given order.

The minimum size of 2-connected chordal bipartite graphs

Abstract

A bipartite graph is chordal bipartite if every cycle of length at least six contains a chord. We determine the minimum size in 2-connected chordal bipartite graphs with given order.

Paper Structure

This paper contains 4 sections, 6 theorems, 5 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proofs of main results
  3. Some remarks
  4. Acknowledgment

Key Result

Theorem 1.1

Let $G$ be a 2-connected chordal bipartite graph of order $n\ge 4$ and size $m$. Then Moreover, the bounds are tight.

Figures (4)

  • Figure 1: Illustration of the structure described in Lemma \ref{['p1']}(5) in the case of $V=S$, where $uv$ is an bisimplicial edge.
  • Figure 2: A chordal bipartite graph with even order $n$ and $\frac{3}{2}n-2$ edges.
  • Figure 3: A chordal bipartite graph with odd order $n$ and $\frac{3}{2}n-\frac{3}{2}$ edges.
  • Figure 4: A chordal bipartite graph with connectivity 3, where $uv$ is a bisimplicial edge.

Theorems & Definitions (15)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2: Golumbic
  • Lemma 2.3: Golumbic
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Claim 2.1
  • proof
  • ...and 5 more