Cohen-Macaulayness of powers of edge ideals of edge-weighted graphs

TL;DR

The paper addresses when the powers of weighted edge ideals $I(G_)$ are Cohen-Macaulay, focusing first on the second power for very well-covered graphs and then on higher powers for special graph classes. It develops a graphependent, inequalityriven framework: for $I(G_)^2$ in weighted bipartite settings, Cohen-Macaulayness is characterized by two families of weight inequalities that relate edge weights across the bipartition; for higher powers, scaled versions of these inequalities yield CM up to a bound $$, with detailed results for trees, stars, and complete-subgraph configurations. The methods combine combinatorial graph structure with commutative algebra tools (Hochster depth, polarization, and primary decompositions) to obtain exact criteria and to establish unmixedness and CM properties of powers. The results enhance understanding of how edge weights and graph topology govern CM behavior of powers of weighted edge ideals, offering concrete criteria and suggesting a broader conjectural link between CM of $I(G_)^2$ and all higher powers.

Abstract

In this paper, we characterize the Cohen-Macaulayness of the second power $I(G_ω)^2$ of the weighted edge ideal $I(G_ω)$ when the underlying graph $G$ is a very well-covered graph. We also characterize the Cohen-Macaulayness of all ordinary powers of $I(G_ω)^n$ when $G$ is a tree with a perfect matching consisting of pendant edges and the induced subgraph $G[V(G)\setminus S]$ of $G$ on $V(G)\setminus S$ is a star, where $S$ is the set of all leaf vertices, or if $G$ is a connected graph with a perfect matching consisting of pendant edges and the induced subgraph $G[V(G)\setminus S]$ of $G$ on $V(G)\setminus S$ is a complete graph and the weight function $ω$ satisfies $ω(e)=1$ for all $e\in E(G[V(G)\setminus S])$.

Figures (5)

• Figure 1: The weighted bipartite graph.
• Figure 2: The weighted path of length $3$.
• Figure 3: The weighted bipartite graph $H_\omega$.
• Figure 4: The graph $G_\omega$ after swapping the endpoints of $x_jy_j$.
• Figure 5: The weighted graph $G_\omega$.

Theorems & Definitions (54)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• proof
• Lemma 2.7
• proof