On the Betti numbers and gracefulness of some planar graphs
  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. 
Maurizio Imbesi and Monica La Barbiera
math.AC