Levelable graphs
Fuente:
arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918170639466496 |
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| author | Bhaskara, Kieran Chong, Michael Y. C. Hibi, Takayuki Ragunathan, Naveena Van Tuyl, Adam |
| author_facet | Bhaskara, Kieran Chong, Michael Y. C. Hibi, Takayuki Ragunathan, Naveena Van Tuyl, Adam |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02065 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Levelable graphs Bhaskara, Kieran Chong, Michael Y. C. Hibi, Takayuki Ragunathan, Naveena Van Tuyl, Adam Combinatorics Commutative Algebra 05C69, 05E40, 13E10 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. |
| title | Levelable graphs |
| topic | Combinatorics Commutative Algebra 05C69, 05E40, 13E10 |
| url | https://arxiv.org/abs/2504.02065 |