Graphs with at most two trees in a forest building process

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1 Grphs with t most two trees in forest uilding process rxiv: v [mth.co] 4 Fe 208 Steve Butler Mis Hmnk Mrie Hrdt Astrct Given grph, we cn form spnning forest y first sorting the edges in some order, nd then only keep edges incident to vertex which is not incident to ny previous edge. The resulting forest is dependent on the ordering of the edges, nd so we cn sk, for exmple, how likely is it for the process to produce grph with k trees. We look t ll grphs which cn produce t most two trees in this process nd determine the proilities of hving either one or two trees. From this we construct infinite fmilies of grphs which re non-isomorphic ut produce the sme proilities. Introduction We consider the following forest uilding process:. Tke ll of the edges of the grph, remove them nd sort them in some order. 2. Go through the edges nd only put those edges ck in which connects to some vertex not previously seen y ny edge. From this, we must end up with forest (or grph without cycles since we cn never dd n edge tht closes cycle. As n exmple, in Figure we list ll 24 different wys to order the edges nd group them sed on the resulting forest formed. We will consider the prolem: How mny different edge orderings produce given numer, sy k, of trees in the resulting grph. Equivlently, wht is the proility tht if we tke rndom ordering of the edges, we produce forest with k trees. We will let P(G,k denote this proility. From Figure, we see tht P(G, 5 nd P(G,2 (note tht 6 6 the proilities need to sum to. This process ws implicitly used in pper of Butler et l. [2] for the complete grph, nd explicitly introduced in pper y Berikkyzy et l. [] where some sic properties were estlished nd the proilities for complete iprtite grphs were determined. We summrize these results here. Deprtment of Mthemtics, Iow Stte University, Ames, IA 500 USA {utler,hmnk,hrdtme}@istte.edu

2 e e 2 e 3 e 4 e e 2 e 3 e 4 e e 2 e 3 e 4 e e 2 e 3 e 4 (e,e 2,e 4,e 3 (e,e 2,e 3,e 4 (e,e 3,e 4,e 2 (e,e 4,e 2,e 3 (e 2,e,e 4,e 3 (e,e 3,e 2,e 4 (e 3,e,e 4,e 2 (e,e 4,e 3,e 2 (e 2,e 4,e,e 3 (e 2,e,e 3,e 4 (e 3,e 4,e,e 2 (e 4,e,e 2,e 3 (e 2,e 4,e 3,e (e 2,e 3,e,e 4 (e 3,e 4,e 2,e (e 4,e,e 3,e 2 (e 4,e 2,e,e 3 (e 2,e 3,e 4,e (e 4,e 3,e,e 2 (e 4,e 2,e 3,e (e 3,e,e 2,e 4 (e 4,e 3,e 2,e (e 3,e 2,e,e 4 (e 3,e 2,e 4,e Figure : The results from different edge orderings of the pw grph. Theorem (Butler et l. [2]. We hve P(K n,k ( n n 2k,k,k ( 2n 2 n 2 n 2k. Theorem 2 (Berikkyzy et l. []. We hve P(K s,t,k (s+t( s t k( k st ( s+t. s For smll grphs (t most 5 vertices, the proilities re given in Berikkyzy et l. []. There re few instnces where two grphs would hve the sme proilities for ll k listed, nd most of those were edge-trnsitive grphs. More generlly, the following ws oserved. Oservtion (Berikkyzy et l. []. If G is n edge-trnsitive grph with minimum degree of t lest 2 nd e is ny edge, then we hve P(G,k P(G e,k for ll k. In essence, this follows y noting tht the lst edge in n ordering is never kept, nd y symmetry every edge is the lst edge in n ordering eqully often. The gol of this note is to compute the proilities for more fmilies of grphs, nmely grphswhich cnproduce tmost two trees in theforest uilding process. Using this, we will produce infinitely mny exmples of non-isomorphic grphs G nd H where the proilities gree nd neither G or H re edge-trnsitive. 2 Grphs with t most two trees We re interested in exploring the grphs which cn produce t most two trees in the forest uilding process. Equivlently, this sttes tht there re t most two disjoint edges in the grph (disjoint in the sense tht they shre no vertex. Proposition. The only non-empty grphs without isolted vertices, which contin no pir of disjoint edges, re str grphs (K,n nd the tringle grph (K 3. Proof. If the grph is not connected, then tking one edge from two different components gives two disjoint edges. So we my ssume the grph is connected. 2

3 Ifthegrphhstwo disjoint edges, thenwecnconnect thesetogether ypth, creting pth of length t lest four. Conversely, if the grph hs pth of length t lest four, then it must contin two disjoint edges. So we cn conclude the longest pth is pth with t most three vertices. If the longest pth hs two vertices, then the grph is K 2. If the longest pth hs three vertices nd the ends of the pth re not lefs then it must e tht the ends connect nd form tringle. Since the pw grph hs two disjoint edges, this cn only hppen if the grph is K 3. Finlly, if we re not tringle nd don t hve pth of length four (nd hence no cycles, then it must e tht we re str. Proposition 2. If grph without isolted vertices hs vertex v of degree t lest five nd contins no set of three disjoint edges, then deleting v nd ll incident edges, nd removing ny isolted vertices results in either n empty grph, str, or K 3. Proof. Let v e vertex of degree t lest 5 in the grph. Suppose tht the resulting grph from deleting the vertex v nd ll incident edges, nd removing ny isolted vertices is not the empty grph, str, or K 3. Proposition sttes tht the only non-empty grphs without isolted vertices, tht contin no set of two disjoint edges, is the str grph nd K 3. Thus the resulting grph will contin t lest two disjoint edges. Cll these edges e nd e 2 ; note neither of these edges re incident to v. At most four edges incident to v re lso incident to the edges e nd e 2. Since v hs degree t lest 5, this leves t lest one edge, e 3, tht is connected to v nd not incident to e or e 2. Thus the originl grph contins set of three disjoint edges. Finlly, we oserve tht if ll the degrees re ounded nd the grph is connected, then s n gets lrge, the dimeter must lso grow which forces three disjoint edges. Puttingthislltogether, weseethtfornlrgeenough(infct, n 6theonlyconnected grphs which produce t most two trees, nd re not strs, re the following five fmilies. GS,,c The strs K,+ nd K,+c which hve leves glued together. (Glued strs. GS +,,c The strs K,+ nd K,+c which hve leves glued together nd the centers joined y n edge. (Glued strs with n edge. Pw The pw grph with leves ppended to the vertex of degree. Di The dimond grph ( four-cycle with n extr edge with leves ppended to one of the vertices of degree 2. (K 4 The complete grph on four vertices with leves ppended to one of the vertices. Note tht the first two of these correspond to Proposition 2 where the remining grph is str; nd the lst three of these correspond to Proposition 2 where the remining grph is K 3. These grphs re shown in Figure 2. 3

4 c c ( GS,,c ( GS +,,c (c Pw (d Di (e (K 4 Figure 2: The five fmilies of grphs. 3 Computing proilities for the fmilies We now turn our ttention to computing the proilities tht grph ends in one or two trees in the forest uilding process. We cn find these proilities y noting tht if there re m edges in the grph, then the proility tht we end with two trees is {rerrngements with two trees} P(G,2. m! We will focus on counting the rerrngements which produce two trees. Prticulrly, we wnt to count rerrngements where t some point n edge occurs, not t the strt, nd involves two vertices which hve not een previously seen. Sincewewillecountingrerrngements, wewillfinditusefultoknowhowtomnipulte inomil coefficients. Recll tht ( n k n! is the numer of wys to choose k elements k!(n k! (in our cse this will usully e loctions out of n n element set. There re mny inomil coefficient identities (see Grhm, Knuth, nd Ptshnik [3, Ch. 5] for good introduction; we will need to mke repeted use of the following well-known result. Proposition 3. We hve ( ( l j q +j m n j ( l+q + m+n+ ( where the sum rnges over ll vlues where the summnds re nonzero. Theorem 3. We hve the following proilities. P(GS,,c, P(GS +,,c, (+c+(+c + (++(+ 2+c+2 (+c+(+c (++( c+ Proof. Since P(GS,,c,+P(GS,,c,2, we cn focus on computing the proility of resulting in two trees. We now clim 4

5 P(GS,,c,2 + c ( ( j i (i+j!(+c i(+2+c i j! (+2+c! ( c ( j i (i+j!(+ i(+2+c i j! (+2+c! +c +2+c + +2+c. (2 This comes from the two cses, nmely where our first edge initilly comes from the top hlf (i.e., edges coming from the str K,+, nd where our first edge initilly comes from the ottom hlf (i.e., edges coming from the str K,+c. We focus on the top hlf cse, s the ottom hlf follows y n identicl rgument y interchnging the roles of nd c. Determining if we hve two trees comes down to wht hppens when we pick our first edge from the str K,+c. We look t ll wys tht this occurs y first picking edges from K,+ nd then considering wht hppens when we pick our edge from K,+c. In prticulr, we will pick j edges from the lef vertices nd i edges from the gluing vertices. We now run over ll possiilities for i nd j. For ech choice of edges we now consider ll possile ordering s follows: ( j corresponds to which of the j edges mong the were chosen. ( i corresponds to which of the i edges mong the were chosen. (i+j! indictes how mny wys to order these i+j edges (note tht these i+j edges re ll of the initil edges. ( + c i indictes how mny edges disjoint from the ones ove re ville to choose, if we wnt to crete two trees. (+2+c i j! is the numer of wys to rerrnge the remining edges. This gives ll orderings of edges possile, to get the proility we now divide y the totl numer of orderings which is (+2+c!. Note tht in the summtion we need to correct for i 0, j 0 which does not fll into the cse where the first edge is from the top. So we sutrct this term off t the end, which gives the +c term t the end. +2+c To now simplify these sums we cn repetedly pply (. So we hve the following. ( ( j i (i+j!(+c i(+2+c i j!!! j!( j! i!( i!!!(+c i (+2+c!( i! (+2+c! (i+j!(+c i(+2+c i j!!!(+c i(2+c i! (+2+c!( i! (+2+c! (i+j!(+2+c i j! i!j! ( j! 5 (i+j!(+2+c i j! i!j! ( j!(2+c i!

6 !!(+c i(2+c i! (+2+c!( i!!!(+c i(2+c i! (+2+c!( i! ( ( i+j +2+c i j i 2+c i ( +2+c 2+c!!(+c i(2+c i! (+2+c! (+2+c!( i! (2+c!!!(+c i(2+c i! ( i!(2+c!!(+c! (2+c! (2+c i! ( i!(+c! (+c i!(+c! ( ( ( 2+c i i ( ( 2+c i i (+c (2+c! +c 0 +c!(+c! ( ( ( 2+c 2+c (+c (2+c! +c +c+!(+c! ( (+c (2+c! (2+c! (+c!! (2+c! (+c+!(! (+c+(+c By similr process, the other doule sum ecomes c ( c ( j i (i+j!(+ i(+2+c i j! (+2+c! (++(+. Now replcing the doule sums y these simplified expressions we hve ( P(GS,,c,2 Finlly we note (+c+(+c P(GS,,c, P(GS,,c,2 +c ( +2+c + (++( c (+c+(+c (++(+. (+c+(+c + (++(+, estlishing the result for GS,,c. The result for P(GS +,,c,2 follows y similr rgument, the only difference eing the dditionl edge which cnnot e used in order to result in two trees. So (2 would now ecome 6

7 P(GS +,,c,2 + ( ( j i (i+j!(+c i(+2+c i j! c ( c ( j i (+2+c+! (i+j!(+ i(+2+c i j! (+2+c+! The rest of the rgument works in the sme wy s efore. Theorem 4. We hve the following proilities. P(Pw, P(Di, P((K 4, c +2+c c+. Proof. We will gin compute the proility tht there re two trees in the process. However, in these cses there re mny more possiilities to consider. To simplify the sitution, we mke the following oservtion: Every edge which is lef in the originl grph will lwys e kept in the forest uilding process. This indictes if there re multiple leves off of single vertex v, then we only need to know when the first lef ws chosen. This is ecuse, y the first lef, v will hve een seen y some edge. So we now represent the remining grphs from Figure 2, s shown in Figure 3, where corresponds to ll of the leves condensed down; nd the remining edges re leled s indicted with ech lel other thn corresponding to single edge. c d d c c d e f e f e g Pw Di (K 4 Figure 3: The remining three grphs with the leves collpsed to single edge. For ech grph, we now look t ll possile wys to strt selecting edges nd end with pir of disjoint edges. We lso find the proility of strting our selection in prticulr wy. Recll tht n edge mrked corresponds to different edges, nd so until we select tht edge, we ssume ll of them hven t een seen nd re ville for picking; fter selection y the oservtion, we cn ssume they hve ll een seen. (In other words, it is only the reltive ordering of the different types of edges tht mtter. For the pw grph, we hve the possiilities shown in Tle (the first column indictes every possile sequence of choices of edges until two trees re formed, while the second 7

8 Strt of edge orderings resulting in 2 trees Proilities of n ordering c, d, e +4 4 e, e c, d, e e e cd, ce, dc, de, ed, ec ced, cde, dce, dec, ecd, edc Tle : Proilities ssocited with Pw. column indictes the proility of ny one of those sequence of choices eing mde. If we now sum ll of these proilities together, we get P(Pw, , estlishing the result (recll tht P(Pw,+P(Pw,2. The results for the remining two grphs re estlished in the sme wy nd the corresponding proilities re given in Tles 2 nd 3. 4 Exmples of grphs with the sme proilities Using the formuls from the theorems in the preceding section, we cn now compute the proilities for lrge numer of these grphs efficiently. In prticulr, we exmined ll grphs up through five hundred vertices in these fmilies, nd discovered severl exmples of fmilies of non-isomorphic grphs which produce the sme proilities. Proposition 4. Given s,t with s dividing into 2t(t+, let r 2t(t+. Then we hve s for ll k P(GS r+3t+,s,t,k P(GS t,r+s+2t+,t,k P(GS 3t+s+,r,t,k. This immeditely follows y pplying the formuls for the proilities from Theorem 3. Proposition 5. Given t, then we hve for ll k P(GS 5t+3,t,2t,k P(GS 5t+,t+,2t+,k. This lso immeditely follows y pplying the formuls for proilities from Theorem 3. We note tht there were mny other exmples of pirs of glued str grphs which re not 8

9 Strt of edge orderings resulting in 2 trees Proilities of n ordering d, e, f +5 5 f, ce, ec, f d, e, f f, ce f, ce de, df, ed, ef, fd, fe def, dfe, edf, efd, fde, fed Tle 2: Proilities ssocited with Di. Strt of edge orderings resulting in 2 trees Proilities of n ordering e, f, g +6 6 f, ce, dg, ec, f, gd e, f, g f, dg, ce f, dg, ce ef, eg, fe, fg, ge, gf efg, egf, feg, fge, gef, gfe Tle 3: Proilities ssocited with (K 4. 9

10 Figure 4: Exmples of set of three grphs with the sme proilities from Proposition 4. In this cse s t. explined y Propositions 4 nd 5. A complete chrcteriztion of ll such pirs of glued strs remins elusive. Looking eyond glued strs, we found very few pirs of grphs with the sme proilities nd the results do not seem to fit ny ptterns. As n exmple, ll pirs of grphs from the GS +,,c fmily up through 500 vertices with the sme proilities re listed elow (it is possile tht this is complete list for this fmily. 5 Conclusion P(GS + 7,3,9,k P(GS+ 0,9,0,k P(GS + 28,5,9,k P(GS+ 26,8,8,k P(GS + 03,5,48,k P(GS+ 63,7,32,k P(GS + 95,23,53,k P(GS+ 53,66,52,k We found the proilities for ll connected grphs which cn form t most two trees in this forest uilding process. A nturl next step is to consider grphs with t most three trees. As n exmple, two pirs of grphs with t most three trees nd mtching proilities re given in Figure 5. This is suggestive tht these re the strt of n infinite fmily of such grphs, ut we hve not yet estlished this. One difficulty is tht unlike the sitution for two trees where only one proility needed to e computed (since the proilities sum to one, this requires tht two proilities e computed. References [] Zhnr Berikkyzy, Steve Butler, Jy Cummings, Kristin Heysse, Pul Horn, Ruth Luo, nd Brent Morn, A forest uilding process on simple grphs, Discrete Mthemtics 34 (208, [2] Steve Butler, Fn Chung, Jy Cummings, nd Ron Grhm, Edge flipping in the complete grph, Advnces in Applied Mthemtics 69 (205, [3] Ronld L. Grhm, Donld E. Knuth, nd Oren Ptshnik, Concrete Mthemtics, second edition, Addison-Wesley,

11 P(G, P(G, P(G, P(G, P(G, P(G, Figure 5: Two pirs of grphs with t most three trees nd producing the sme proilities.