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Algebraic and geometric aspects of bipartite planar graphs

Maurizio Imbesi, Monica La Barbiera

TL;DR

The bounds for the graded Betti numbers and the projective dimension of the quotient ring associated to the graph are found and the minimal vertex covers and the maximum matchings related to such a graph are investigated.

Abstract

Let B_{2t} be a bipartite planar graph with an even number of regions. We are able to find bounds for the graded Betti numbers and the projective dimension of the quotient ring associated to the graph. We also will investigate the minimal vertex covers and the maximum matchings related to such a graph.

Algebraic and geometric aspects of bipartite planar graphs

TL;DR

The bounds for the graded Betti numbers and the projective dimension of the quotient ring associated to the graph are found and the minimal vertex covers and the maximum matchings related to such a graph are investigated.

Abstract

Let B_{2t} be a bipartite planar graph with an even number of regions. We are able to find bounds for the graded Betti numbers and the projective dimension of the quotient ring associated to the graph. We also will investigate the minimal vertex covers and the maximum matchings related to such a graph.

Paper Structure

This paper contains 4 sections, 15 theorems, 16 equations, 7 figures.

Table of Contents

  1. Preliminary notions
  2. Graded Betti numbers associated to $B_{2t}$
  3. Algebraic topics on minimal vertex covers of $B_{2t}$
  4. Perfect matchings of $B_{2t}$

Key Result

Theorem 1.2

A graph is planar if and only if it has no subgraphs containing $K_5$ and $K_{3,3}$.

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.2: H, Theorem 11.13
  • Example 1.3
  • Definition 1.4
  • Definition 2.1
  • Proposition 2.2: J, 4.1.1 Proposition
  • Proposition 2.3
  • proof
  • Theorem 2.4: El:Vill
  • Theorem 2.5
  • ...and 26 more