Collectanea Mathematica

, Volume 69, Issue 2, pp 249–262

# Improved bounds for the regularity of edge ideals of graphs

Article

## Abstract

Let G be a graph with n vertices, let $$S={\mathbb {K}}[x_1,\dots ,x_n]$$ be the polynomial ring in n variables over a field $${\mathbb {K}}$$ and let I(G) denote the edge ideal of G. For every collection $${\mathcal {H}}$$ of connected graphs with $$K_2\in {\mathcal {H}}$$, we introduce the notions of $${{\mathrm{ind-match}}}_{{\mathcal {H}}}(G)$$ and $${{\mathrm{min-match}}}_{{\mathcal {H}}}(G)$$. It will be proved that the inequalities $${{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)\le \mathrm{reg}(S/I(G))\le {{\mathrm{min-match}}}_{\{K_2, C_5\}}(G)$$ are true. Moreover, we show that if G is a Cohen–Macaulay graph with girth at least five, then $$\mathrm{reg}(S/I(G))={{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)$$. Furthermore, we prove that if G is a paw-free and doubly Cohen–Macaulay graph, then $$\mathrm{reg}(S/I(G))={{\mathrm{ind-match}}}_{\{K_2, C_5\}}(G)$$ if and only if every connected component of G is either a complete graph or a 5-cycle graph. Among other results, we show that for every doubly Cohen–Macaulay simplicial complex, the equality $$\mathrm{reg}({\mathbb {K}}[\Delta ])=\mathrm{dim}({\mathbb {K}}[\Delta ])$$ holds.

## Keywords

Edge ideal Castelnuovo–Mumford regularity Girth Matching Paw-free graph

## Mathematics Subject Classification

Primary: 13D02 05E99 Secondary: 13F55

## Notes

### Acknowledgements

The authors thank the referee for careful reading of the paper and for useful comments.

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