On the Betti numbers and gracefulness of some planar graphs

TL;DR

This paper explores the algebraic properties of bipartite planar graphs with a focus on Betti numbers and projective dimension, establishing connections between graph regions and algebraic invariants, and demonstrating graceful labelings.

Contribution

It introduces bounds for algebraic invariants of bipartite planar graphs and proves their relation to the number of regions, along with a method for graceful labeling.

Findings

Bounds for graded Betti numbers are established.

A relationship between the number of regions and algebraic invariants is demonstrated.

Graphs St_r admit graceful edge labelings.

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.