## 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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## References

- [1]M. Adamaszek: Splittings of independence complexes and the powers of cycles,
*Journal of Combinatorial Theory Series A***119**(2012), 1031–1047.MathSciNetCrossRefzbMATHGoogle Scholar - [2]D. Attali, A. Lieutier and D. Salinas: Effcient data structure for representing and simplifying simplicial complexes in high dimensions,
*International Journal of Computational Geometry and Applications***22**(2012), 279–303.MathSciNetCrossRefzbMATHGoogle Scholar - [3]T. Biyikoğlu and Y. Civan: Four-cycled graphs with topological applications,
*Annals of Combinatorics***16**(2012), 37–56.MathSciNetCrossRefzbMATHGoogle Scholar - [4]T. Biyikoğlu and Y. Civan: Bounding Castelnuovo-Mumford regularity of graphs via Lozin's operations, unpublished manuscript, available at arXiv:1302.3064, 2013.Google Scholar
- [5]T. Biyikoğlu and Y. Civan: Vertex-decomposable graphs,
*codismantlability, Cohen-Macaulayness, and Castelnuovo-Mumford regularity, Electronic Journal of Combinatorics,***21**(1):#P1, 2014.Google Scholar - [6]T. Biyikoğlu and Y. Civan: Castelnuovo-Mumford regularity of graphs, available at arXiv:1503.06018(v1), 43pp, 2015.zbMATHGoogle Scholar
- [7]A. Brandstädt, V. B. Le and J. P. Spinrad: Graph Classes, A Survey, SIAM Monographs on Discrete Mathematics and Applications, Philadelphia, 1999.CrossRefGoogle Scholar
- [8]P. Csorba: Subdivision yields Alexander duality on independence complexes,
*Electronic Journal of Combinatorics,***16**(2):#R11, 2009.Google Scholar - [9]H. Dao, C. Huneke and J. Schweig: Bounds on the regularity and projective dimension of ideals associated to graphs,
*Journal of Algebraic Combinatorics***38**(2013), 37–55.MathSciNetCrossRefzbMATHGoogle Scholar - [10]R. Ehrenborg and G. Hetyei: The topology of the independence complex,
*European Journal of Combinatorics***27**(2006), 906–923.MathSciNetCrossRefzbMATHGoogle Scholar - [11]A. Engström: Complexes of directed trees and independence complexes,
*Discrete Mathematics***309**(2009), 3299–3309.MathSciNetCrossRefzbMATHGoogle Scholar - [12]
- [13]H. T. Hà: Regularity of squarefree monomial ideals, in: S. M. Cooper and S. Sather-Wagstaff, editors,
*Connections Between Algebra, Combinatorics, and Geometry*, volume 76, 251–276. Springer, Proceedings in Mathematics and Statistics, 2014.MathSciNetCrossRefzbMATHGoogle Scholar - [14]H. T. Hà and A. V. Tuyl: Monomial ideals,
*edge ideals of hypergraphs, and their graded betti numbers, Journal of Algebraic Combinatorics***27**(2008), 215–245.MathSciNetCrossRefzbMATHGoogle Scholar - [15]T. Januszkiewicz and J. Świątkowski: Hyperbolic Coxeter groups of large dimension,
*Commentarii Mathematici Helvetici***78**(2003), 555–583.MathSciNetCrossRefzbMATHGoogle Scholar - [16]G. Kalai and R. Meshulam: Intersection of Leray complexes and regularity of monomial ideals,
*Journal of Combinatorial Theory Series A***113**(2006), 1586–1592.MathSciNetCrossRefzbMATHGoogle Scholar - [17]M. Katzman: Characteristic-independence of Betti numbers of graph ideals,
*Journal of Combinatorial Theory Series A***113**(2006), 435–454.MathSciNetCrossRefzbMATHGoogle Scholar - [18]D. Kozlov:
*Combinatorial Algebraic Topology*, volume ACM 21, Springer, Berlin, 2008.CrossRefzbMATHGoogle Scholar - [19]V. V. Lozin: On maximum induced matchings in bipartite graphs,
*Information Processing Letters***81**(2002), 7–11.MathSciNetCrossRefzbMATHGoogle Scholar - [20]F. H. Lutz and E. Nevo: Stellar theory for flag complexes,
*Mathematica Scandinavica***118**(2016), 70–82.MathSciNetCrossRefzbMATHGoogle Scholar - [21]M. Mahmoudi, A. Mousivand, M. Crupi, G. Rinaldo, N. Terai and S. Yassemi: Vertex decomposability and regularity of very wellcovered graphs,
*Journal of Pure and Applied Algebra***215**(2011), 2473–2480.MathSciNetCrossRefzbMATHGoogle Scholar - [22]M. Marietti and D. Testa: A uniform approach to complexes arising from forests,
*Electronic Journal of Combinatorics,***15**:#R101, 2008.Google Scholar - [23]D. Marušič and T. Pisanski: The remarkable generalized Petersen graph G(8; 3),
*Mathematica Slovaca***50**(2000), 117–121.MathSciNetzbMATHGoogle Scholar - [24]S. Morey and R. H. Villarreal: Edge ideals: algebraic and combinatorial properties, in: C. Francisco, L. C. Klingler, S. Sather-Wagstaff, and J. C. Vassilev, editors,
*Progress in Commutative Algebra 1: Combinatorics and Homology*, chapter 3, 85–126. De Gruyter, Berlin, 2012.Google Scholar - [25]E. Nevo: Regularity of edge ideals of C4-free graphs via the topology of the lcm-lattice,
*Journal of Combinatorial Theory Series A***118**(2011), 491–501.MathSciNetCrossRefzbMATHGoogle Scholar - [26]E. Nevo and I. Peeva: C4-free edge ideals,
*Journal of Algebraic Combinatorics***37**(2013), 243–248.CrossRefzbMATHGoogle Scholar - [27]D. Osajda: A construction of hyperbolic Coxeter groups,
*Commentarii Mathematici Helvetici***88**(2013), 353–367.MathSciNetCrossRefzbMATHGoogle Scholar - [28]P. Przytycki and J. Świątkowski: Flag-no-square triangulations and Gromov boundaries in dimension 3,
*Groups, Geometry, and Dynamics***3**(2013), 453–468.MathSciNetzbMATHGoogle Scholar - [29]R. P. Stanley:
*Combinatorics and Commutative Algebra, Second Edition*, volume 41, Progress in Mathematics, Birkhäuser, Boston, MA, 1996.Google Scholar - [30]W. A. Stein et al:
*Sage Mathematics Software*, The Sage Development Team, http://www.sagemath.org, 2014.Google Scholar - [31]A. V. Tuyl: Sequentially Cohen-Macaulay bipartite graphs: vertex decomposability and regularity,
*Archiv der Mathematik***93**(2009), 451–459.MathSciNetCrossRefzbMATHGoogle Scholar - [32]D. W. Walkup: The lower bound conjecture for 3 and 4-manifolds,
*Acta Mathematica***125**(1970), 75–107.MathSciNetCrossRefzbMATHGoogle Scholar - [33]G. Weetman: A construction of locally homogeneous graphs,
*Journal of the London Mathematical Society***50**(1994), 68–86.MathSciNetCrossRefzbMATHGoogle Scholar - [34]G. Whieldon: Jump sequences of ideals, preprint, available at arXiv:1012.0108v1, 27pp, 2015.Google Scholar
- [35]R. Woodroofe: Matchings, coverings, and Castelnuovo-Mumford regularity,
*Journal of Commutative Algebra***6**(2014), 287–304.MathSciNetCrossRefzbMATHGoogle Scholar