EDGE AND SECANT IDEALS OF SHARED-VERTEX GRAPHS

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1 EDGE AND SECANT IDEALS OF SHARED-VERTEX GRAPHS ZVI ROSEN Abstract. We examie miimal free resolutios ad Betti diagrams of the edge ad secat ideals of oe family of graphs. We demostrate how splittig the graph explais patters observed i the Betti diagrams. 1. Itroductio We will focus o a family of graphs that we will call shared-vertex graphs, which cosist of distict triagles joied together at oe vertex. Each graph correspods to a polyomial rig cotaiig variables correspodig to each vertex. Whe we discuss the edge ideals of a graph, we are referrig to the ideal geerated by the degree-two moomials that correspod to the edges of the graph. For istace, the edge AB correspods to the moomial ab. Figure 1. A Graph ad Its Correspodig Edge Ideal The miimal free resolutio of a give edge ideal, will refer to the exact sequece defied by its fial homomorphism: multiplicatio by the geerators of our ideal. The Betti diagram summarizes the umber of iputs ad degree at each step of the resolutio. Supported by NSF grat DMS

2 2 ZVI ROSEN Use of the Macaulay 2 program has demostrated a predictable patter i the Betti diagrams of the edge ad secat ideals of the sharedvertex graphs. I 2 we demostrate the patter i the Betti Diagrams. I 3 we demostrate a coveiet splittig for our family of graphs. I 4 we show how the splittig explais the patter i the Betti diagrams. 2. Betti Diagrams After examiig the cases startig with two triagles joied at a vertex, ad goig up to 6 triagles joied at a cetral vertex, the Betti Diagrams follow a very predictable patter. Let be the umber of triagles i the graph. The the Betti Diagram for the edge ideal will be of the followig form: ) ) 4 3) ) 1 ) Lookig at this Betti diagram, three patters are obvious: The top row, which gives the biomial coefficiets of 2. The two diagoals, cotaiig the biomial coefficiets of. There is overlap amog these three patters i the first two colums of the top row. As for the secat ideal, the Betti Diagram is eve simpler: ) 4 2) ) Here we oly have oe diagoal patter of the biomial coefficiets up to. I the remaiig sectios, we will show that these patters will persist for all values of.

3 EDGE AND SECANT IDEALS OF SHARED-VERTEX GRAPHS 3 3. Splittig the Graph To try ad make sese of these patters, let s look agai at oe example of a shared-vertex graph. Below is the graph with = 3 triagles. Figure 2. 3-Triagle Graph You may otice that you ca divide the edges ito two categories: 1) The 2 edges aliged radially from the ceter, which we will call r-edges. 2) The edges aliged alog the perimeter, which we will call p-edges. Figure 3. Highlighted r-edges ad p-edges This distictio leds itself to a coveiet splittig o our graph, a techique described i [1], ad paraphrased below. Defiitio 3.1. Eliahou-Kervaire) A moomial ideal I is splittable if I is the sum of two ozero moomial ideals J ad K, that is, I = J + K, such that

4 4 ZVI ROSEN 1) The miimal geeratig set GI) is the disjoit uio of GJ) ad GK), ad 2) there is a splittig fuctio GJ K) GJ) GK) w φw),ψw)) such that: 1) for all w GJ K), w = lcmφw),ψw)). 2) for every subset S GJ K), both lcmφs)) ad lcmψs)) strictly divide lcms) Give this defiitio, we will split up our shared vertex graphs ito the set of r-edges ad the set of p-edges, as is show i the images below. Figure 4. The subgraphs correspodig to J ad K Switchig ow to the polyomial rig k[x 0,..., x 2 ] for coveiece, we ca describe the edge ideal i the triagle case, ad its compoet ideals as follows: 1) I =< x 0 x 1,x 0 x 2,x 1 x 2,..., x 0 x 2 1,x 0 x 2,x 2 1 x 2 >, the etire edge ideal. 2) J =< x 0 x 1,x 0 x 2,..., x 0 x 2 1,x 0 x 2 >= x 0 <x 1,x 2,..., x 2 >, the ideal geerated by the r-edges. 3) K =< x 1 x 2,x 3 x 4,..., x 2 1 x 2 >, the ideal geerated by the p-edges. 4) J K =< x 0 x 1 x 2,x 0 x 3 x 4,..., x 0 x 2 1 x 2 >, the itersectio of the ideals which icidetally coicides with our secat ideal), or the product of the variables represetig the vertices of each triagle.

5 EDGE AND SECANT IDEALS OF SHARED-VERTEX GRAPHS 5 We ca defie the splittig fuctio to take oe of the degree-3 moomials i the itersectio to two degree-2 moomials : GJ K) GJ) GK) w w gcdw, x 2 x 4...x 2 ), w ) x 0 For istace, take ay triagle i the itersectio J K, which will have geeral form x 0 x 2i 1 x 2i. The φ fuctio will get rid of x 2i, leavig x 0 x 2i 1, a r-edge; the ψ fuctio gets rid of the x 0, leavig x 2i 1 x 2i, a p-edge. The required properties for a splittig fuctio are easily see to be satisfied. Note that ay similar product of ucoected vertices would work for the gcd i the φ fuctio.) 4. Compoets of the Betti Diagram Havig established that our edge ideal is splittable, we ca utilize a powerful relatio about graded Betti umbers established by Fattabi, also reported i [1]. Theorem 4.1. Suppose I is a splittable moomial ideal with splittig I = J + K. The β i,j I) =β i,j J)+β i,j K)+β i 1,j J K) for all i, j 0 where β i 1,j J K) = 0 if i =0. Now, we have to look oly at the much simpler ideals J, K, ad J K, i order to describe the Betti diagram of I. First, let s look at J. I the triagle case, this ideal is geerated by 2 quadratic moomials, each cotaiig a factor of x 0 ad aother x i uique to that moomial. Sice x 0 is i every term, it will ot appear i ay of the syzygies, so we ca cosider this a ideal of degree-1 moomials x 1,.., x 2. This is a Koszul complex, whose Betti diagram is of the form as described i [2]): ) 2 The ideal K is geerated by the quadratic moomials correspodig to the outer edges. Noe of these edges share ay vertices, thus the moomials are all relatively prime. This too forms a slightly modified

6 6 ZVI ROSEN Koszul complex. I this complex, each syzygy will have oly degree- 2 terms, sice for ay two moomials m i ad m j, m i s coefficiet i lcmm the syzygy, i.e. i,m j ) m j, will simply be the other quadratic moomial. So, the Betti diagram will look the same, except each step i the resolutio will be oe degree higher tha the previous oe, as i the followig diagram ) 3 2) ) As we metioed previously, the ideal J K is equivalet to the secat ideal, whose Betti diagram appeared earlier. We ca explai this patter by otig that, as i J, every term has a x 0 factor that ca be disregarded; ad as i K, each term has two factors ushared by the other terms. Therefore, we oce agai have a Koszul complex, with degree-2 moomials. Usig Kattabi s relatio for the graded Betti umbers, we start by simply addig the Betti diagrams for J ad K together. The J K terms are shifted over oe colum, because we take the β i-1,j J K) i the β i,j I). The, they are shifted up oe row as well, because the rows are idexed accordig to the colum. What we are left with is the Betti diagram show i 1.

7 EDGE AND SECANT IDEALS OF SHARED-VERTEX GRAPHS 7 Refereces [1] Huy Tài Hà ad Adam Va Tuyl. Moomial Ideals, Edge Ideals of Hypergraphs, ad their Miimal Graded Free Resolutios. Joural of Algebraic Combiatorics: A Iteratioal Joural Volume 27, Issue 2. April 2008) Pages [2] David Eisebud, with a chapter by Jessica Sidma. Lectures o the Geometry of Syzygies. Treds i Commutative Algebra Volume ) Pages [3] Grayso, Daiel R. ad Stillma, Michael E., Macaulay 2, a software system for research i algebraic geometry, Available at