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On the Betti numbers and gracefulness of some planar graphs

Maurizio Imbesi, Monica La Barbiera

Abstract

In this article bipartite planar graphs St_r are investigated, r the number of their plane regions. Bounds for the graded Betti numbers and the projective dimension of the quotient ring associated to such graphs are discussed. We prove that r is related to algebraic invariants that arise from the projective resolution of the edge ideal of the graph. We also deal with labeling methods for certain graphs and show that graphs St_r admit a graceful numbering of their edges.

On the Betti numbers and gracefulness of some planar graphs

Abstract

In this article bipartite planar graphs St_r are investigated, r the number of their plane regions. Bounds for the graded Betti numbers and the projective dimension of the quotient ring associated to such graphs are discussed. We prove that r is related to algebraic invariants that arise from the projective resolution of the edge ideal of the graph. We also deal with labeling methods for certain graphs and show that graphs St_r admit a graceful numbering of their edges.

Paper Structure

This paper contains 3 sections, 13 theorems, 11 equations, 3 figures.

Table of Contents

  1. Preliminary notions
  2. Syzygy modules of $I(St_r)$
  3. Graceful labeling of graphs $St_r$

Key Result

Theorem 1.5

A graph is planar if and only if it has no subgraph homeomorphic to $K_5$ or $K_{3,3}$.

Figures (3)

  • Figure 1: Two representations of the graph $St_2$
  • Figure 2: A graceful labeling of $St_6$ or $G_6$
  • Figure 3: A graceful labeling of $G_7$ or $J_{2,7}$

Theorems & Definitions (41)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5: Kuratowski
  • proof
  • Remark 1.6
  • Remark 1.7
  • Example 1.8
  • Theorem 2.1
  • ...and 31 more