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arXiv · 2504.02065

Levelable graphs

Kieran Bhaskara·Michael Y. C. Chong·Takayuki Hibi·Naveena Ragunathan·Adam Van Tuyl
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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.

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Kieran Bhaskara, Michael Y. C. Chong, Takayuki Hibi, Naveena Ragunathan, Adam Van Tuyl. 2025-04-02. Levelable graphs. https://arxiv.org/abs/2504.02065

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