Levelable graphs
Kieran Bhaskara, Michael Y. C. Chong, Takayuki Hibi, Naveena Ragunathan, Adam Van Tuyl
TL;DR
The paper introduces levelable graphs, a weighted generalization of well-covered graphs, and links them to level artinian rings via edge ideals. It develops constructions to generate levelable graphs from known ones and classifies levelable instances in several families, including Cameron–Walker graphs, chordal/co-chordal graphs, and circulant graphs. A key contribution is the precise algebraic bridge: G is levelable with weights c iff $R/I(G)+\langle x_i^{c_i+1}\rangle$ is a level graded artinian ring, enabling transfer of combinatorial results to algebraic consequences. The authors also show that, thanks to Brown–Nowakowski, almost all graphs fail to be levelable and, consequently, that the associated graded rings are typically not level or Cohen–Macaulay.
Abstract
We study a family of positive weighted well-covered graphs, which we call levelable graphs, that are related to a construction of level artinian rings in commutative algebra. A graph $G$ is levelable if there exists a weight function with positive integer values on the vertices of $G$ such that $G$ is well-covered with respect to this weight function. That is, the sum of the weights in any maximal independent set of vertices of $G$ is the same. We describe some of the basic properties of levelable graphs and classify the levelable graphs for some families of graphs, e.g., trees, cubic circulants, Cameron--Walker graphs. We also explain the connection between levelable graphs and a class of level artinian rings. Applying a result of Brown and Nowakowski about weighted well-covered graphs, we show that for most graphs, their edge ideals are not Cohen--Macaulay.