2025/04/02 by Bhaskara, Kieran, Chong, Michael Y. C., Hibi, Takayuki +2
#05C69 #05E40 #13E10 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2504.02065
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.