# Castelnuovo-Mumford Regularity of Graphs

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## Abstract

We present new combinatorial results on the calculation of (Castelnuovo-Mumford) regularity of graphs. We introduce the notion of a *prime graph* over a field k, which we define to be a connected graph with reg_{k}(*G* − *x*) < reg_{k}(*G*) for any vertex *x* ∈ *V* (*G*). We then exhibit some structural properties of prime graphs. This enables us to provide upper bounds to the regularity involving the induced matching number im(*G*). We prove that reg(*G*) ≤ (*Γ*(*G*)+1)im(*G*) holds for any graph *G*, where *Γ*(*G*)=max{|*N*_{ G }[*x*]\*N*_{ G }[*y*]| : *xy* ∈ *E*(*G*)} is the *maximum privacy degree* of *G* and *N*_{ G }[*x*] is the closed neighbourhood of *x* in *G*. In the case of claw-free graphs, we verify that this bound can be strengthened by showing that reg(*G*)≤2im(*G*). By analysing the effect of Lozin transformations on graphs, we narrow the search for prime graphs into graphs having maximum degree at most three. We show that the regularity of such graphs *G* is bounded above by 2im(*G*)+1. Moreover, we prove that any non-trivial Lozin operation preserves the primeness of a graph. That enables us to generate many new prime graphs from the existing ones.

We prove that the inequality reg(*G*/*e*)≤reg(*G*)≤reg(*G*/*e*)+1 holds for the contraction of any edge *e* of a graph *G*. This implies that reg(*H*) ≤ reg(*G*) whenever *H* is an edge contraction minor of *G*.

Finally, we show that there exist connected graphs satisfying reg(*G*)=*n* and im(*G*)=*k* for any two integers *n* ≥ *k* ≥ 1. The proof is based on a result of Januszkiewicz and Świa̦tkowski on the existence of Gromov hyperbolic right angled Coxeter groups of arbitrarily large virtual cohomological dimension, accompanied with Lozin operations. In an opposite direction, we show that if *G* is a 2*K*_{2}-free prime graph, then reg(*G*)≤(*δ*(*G*)+3)/2, where *δ*(*G*) is the minimum degree of *G*.

## Mathematics Subject Classification (2000)

13F55 05E40 05C70 05C75 05C76## Preview

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