Algebraic and geometric aspects of bipartite planar graphs

TL;DR

This paper explores algebraic and geometric properties of bipartite planar graphs, establishing bounds on algebraic invariants and analyzing combinatorial features like vertex covers and matchings.

Contribution

It provides new bounds for Betti numbers and projective dimension in relation to bipartite planar graphs with even regions, linking algebraic and combinatorial aspects.

Findings

Bounds for graded Betti numbers established

Bounds for projective dimension derived

Analysis of minimal vertex covers and maximum matchings

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.