Symmetric Class 0 subgraphs of complete graphs

Transcription

1 DIMACS Techical Report 0-0 November 0 Symmetric Class 0 subgraphs of complete graphs Vi de Silva Departmet of Mathematics Pomoa College Claremot, CA, USA Chaig Verbec, Jr. Becer Friedma Istitute Booth School of Busiess Uiversity of Chicago Chicago, IL, USA November 0 Eugee Fiorii DIMACS Rutgers Uiversity Piscataway, NJ, USA Abstract I graph pebblig, a coected graph is called Class 0 if it has a pebblig umber equal to its umber of vertices. This paper addresses the questio of whe it is possible to edge-partitio a complete graph ito complemetary Class 0 subgraphs. We defie the otio of -Class 0 graphs: a graph G is -Class 0 if it cotais edge-disjoit subgraphs, where each subgraph is Class 0. We ext preset a family of -Class 0 graphs for =, showig that for 9, K is -Class 0. We fially provide a probabilistic argumet to prove that N, N such that K ca be edge-partitioed ito cyclically symmetric subgraphs of diameter ad coectivity 3: that is K is -Class 0.

2 Itroductio I the traditioal pebblig framewor for a graph G, we defie a pebblig distributio P : V(G) N which places a umber of pebbles o each vertex of the graph[]. A legal pebblig move cosists of taig two pebbles from oe vertex ad removig them from the graph, while addig oe pebble to a adjacet vertex. May problems i graph pebblig relate to the pebblig umber π(g) of the graph (the miimum umber of pebbles required such that ay pebblig distributio placig that may pebbles o the graph will allow for ay target vertex to be reached through a series of pebblig moves). I this paper we will study the partitioig of complete graphs ito complimetary cyclically symmetric Class 0 subgraphs. We defie the otio of -Class 0 graphs: a graph G is -Class 0 if it cotais edge-disjoit subgraphs, where each subgraph is Class 0. To provide cotext, oe ca cosider a multi-colored relative of the classic graph pebblig problem. The game i questio is a geeralizatio of graph pebblig to a versio with players, each represetig oe of colors which are assiged to both pebbles ad edges (see [] for a similar game). Whe ca the edges of a graph be colored with colors, such that each implies a Class 0 subgraph? We show that for a give atural, there exists a such that the complete graph o vertices ca be edge-partitioed ito circularly symmetric Class 0 subgraphs, ad hece is -Class 0. We ll begi by itroducig defiitios relevat to -Class 0 graphs, as well as a ow sufficiet coditio for a graph to be Class 0: diameter ad 3-coectivity (D3C). We preset a set of costructive operatios o D3C graphs, ad fid a family of -Class 0 graphs. We the itroduce a costructio for cyclically symmetric subgraphs of complete graphs. We fially preset a probabilistic argumet to prove our mai result. Defiitios ad Framewor Recall the followig importat defiitios from stadard graph pebblig [3]: Defiitio. A pebblig move cosists of the removal of two pebbles from a sigle vertex ad addig oe to a adjacet vertex. Defiitio. The pebblig umber for a graph G, π(g), is the miimum umber such that ay distributio of π(g) pebbles o G will allow for ay target vertex to be reached by a series of pebblig moves. It is clear that π(g) V (G). Defiitio. A graph G is Class 0 if π(g) = V (G). We itroduce the followig pertiet defiitio cocerig Class 0 subgraphs of a Class 0 graph. Defiitio. A graph G o vertices is cosidered -Class 0 if it cotais edge-disjoit Class 0 subgraphs spaig vertices.

3 A graph which is -Class 0 ca hece be partitioed ito subgraphs, leavig room for a -player graph pebblig game with each player movig o a Class 0 subgraph of the iitial graph. It is clear that beig -Class 0 is a mootoe property i graph edges; that is, addig edges to a graph which is already -Class 0 will leave it -Class 0. Note also that these subgraphs do ot eed to be a precise decompositio; uassiged edges wo t affect -Class 0 by mootoicity. Two properties of graphs which are relevat i the discussio of Class 0 graphs are diameter ad coectivity. Defiitio. Let G be a coected graph. The the diameter of G, diam(g), is the greatest distace betwee ay pair of vertices. Defiitio. A graph G graph is said to be -coected if there does ot exist a set of vertices whose removal discoects the graph. By covetio, we require G to either have + or greater vertices, or be the complete graph o + vertices. For a give graph, the maximum such that the graph is -coected is writte as κ(g), the graph s vertex coectivity. Ituititively, oe ca see that graphs of smaller diameter are more liely to be Class 0, ad that low coectivity prevet a graph from beig Class 0. There has bee much wor doe characterizig Class 0 graphs, but oe result which is used multiple times i our paper is give here. Theorem (Clare, et. al.). If diam(g) =, ad κ(g) 3, the G is of Class 0 [4]. It will be coveiet to defie this subset of Class 0 graphs, as results i this paper wor directly with diameter ad coectivity as a sufficiet coditio for Class 0. Defiitio. We call a graph D3C if it has diameter ad is 3-coected. From the previous theorem, we see that the D3C coditio is stroger tha that of Class 0. I particular, this coditio implies that almost all graphs are Class 0, sice almost all graphs are D3C [4]. While D3C is a sufficiet coditio for beig Class 0, it is ot ecessary; a subsequet theorem [5] states that there exists a fuctio (d), bouded i d (d) d+3, such that if G is a graph of diameter d, ad κ(g) d (d), the G is of Class 0. Below we preset a example of a Class 0 graph with high coectivity but diameter greater tha. Example. Cosider the icosahedro, a graph with 5-regular vertices, alteratively defied as havig 0 equilateral triagle faces. Oe ca show that the icosahedro has diameter 3 ad is 5-coected, but is Class 0. Note that the coectivity does ot lie withi the bouds of the fuctio (d) guarateed by the theorem metioed above [5]. 3 Operatios o D3C Graphs We ow preset a pair of operatios o graphs which preserve D3C. 3

4 Defiitio (Strog Product). Give graphs G ad H, we ll defie G H as a graph with vertex set equal to the Cartesia Product of V (G) ad V (H), ad a edge relatio such that (u i, v i )(u j, v j ) E(G H) if ay oe of the followig holds: (i) u i u j E(G) ad v i = v j (ii) u i = u j ad v i v j E(H) (iii) u i u j E(G) ad v i v j E(H) We call this the Strog Product of the two graphs. Defiitio (Vertex Splittig). Let G be a graph o vertices, ad v V (G). We split the vertex v to form graph G v, duplicatig the vertex by addig v 0, ad coectig v 0 by a edge to all adjacecies of v. Note that G v is ow a graph o + vertices. I particular, we fid that both strog products ad vertex splittig preserve D3C across ay umber of subgraphs. Theorem. Let G ad H be graphs which ca be edge-partitioed ito D3C graphs. The G H ca be edge-partitioed ito D3C graphs, ad hece is -Class 0. Theorem. Let G be a graph which ca be edge-partitioed ito D3C graphs. The for ay v V (G), G v ca be edge-partitioed ito D3C graphs, ad hece is -Class 0. Sice we ca show that K 9 is -Class 0 by example (see Figure ), a cosequece of Theorem is our first family of -Class 0 graphs. Theorem 3. For 9, complete graphs of the form K ca be partitioed ito two D3C subgraphs, ad i particular are -Class 0. 4 Cyclic Costructios While the previous sectio provides a family of -Class 0 graphs ad two costructive operatios o D3C graphs, the ext will itroduce a cocrete costructio for this family whe is odd. This cyclically symmetric costructio will lay the foudatio for the existece result i the fial sectio. Defiitio (Cyclic Costructio). Let N, S = {,..., }. Let A = {a, a,...} S, ad defie C (A) = C (a, a,...) as the graph o vertices v, v,..., v, where v i v j E(C (A)) if ad oly if j i = ±m mod for some m A. Remar. Note that C (A) is symmetric uder cyclic permutatios of v,..., v, ad is a subgraph of K. Note also that it suffices to cosider S with cardiality, as paths to the vertices of the opposite half will follow by symmetry. Remar. Partitioig the edges of the graph ito cyclically symmetric subgraphs is equivalet to partitioig the set S = {,,..., } ito subsets A,..., A such that the elemets of a give subset will represet the edge legths uiquely attributed to that subgraph accordig to our cyclic costructio. Moreover, it is clear that K = C (S) = C (A ) C (A ). 4

5 G = C 9 (, ) G c = C 9 (3, 4) K 9 Figure : -Class 0 costructio for = 9 Example. We ow focus o the particular subgraph costructio that will be relevat i the ext proof. Let r {, 3,...}, ad cosider G = C 4r+ (,..., r) ad G c = C 4r+ (r +,..., r). We more clearly illustrate this costructio whe r =, = 9 i Figure. I Figure, S = {,, 3, 4}, where A = {, } ad A = {3, 4} defie C 9 (, ) ad C 9 (3, 4). Note also that K 9 = C 9 (, ) C 9 (3, 4). C (A) i geeral appears to be highly coected. The followig lemma gives a example of whe 3-coectivity is guarateed. Lemma 4. Let N, A S = {,..., }, ad cosider C (A). If A cotais a elemet s such that gcd(s, ) =, ad at least oe other elemet, the C (A) is 3-coected. Proof. Let s, t S such that s ad are coprime, ad cosider C(s, t). Startig with a particular vertex v 0, we relabel the vertices v 0, v, v,..., v as v 0, v s, v s,..., v ( )s, to show that C (s, t) C (, s t). From a arbitrary -vertex cut, we are left with at most two coected compoets from the circular path of edge-legth aloe (see Figure ). Oe of the resultig arcs must have less tha or equal to the umber of vertices i the other, ad we label the vertices circularly such that we removed v 0, v for. Sice s t ad the arc cotaiig our vertex is smaller, we have + s t <. This implies that a edge coects v to a vertex (ot v 0 ) i the opposite arc, from which we coclude that the two arcs are coected, ad the graph itself is 3-coected. We surmise that the actual coectivity coditio is stroger, but this lemma suffices. By restrictig ourself to a prime umber of vertices i a complete graph, ay subpartitio with two or more edge legths will be 3-coected. Theorem 5. Complete graphs of the form K 4r+ for r =, 3,... ca be edge-partitioed ito two cyclically symmetric D3C subgraphs C 4r+ (,..., r) ad C 4r+ (r +,..., r), ad hece are -Class 0. 5

6 K = C (S) C () C () \ {v, v 8 } C (, 3) \ {v, v 8 } Figure : Illustratio of proof method of Theorem 5 where =, s =, t = 3. It is clear that as log as s t, the two compoets must be coected by a elemet of the subgraph geerated by t. Proof. Let r {, 3, }, ad let H = K 4r+. We ow show that usig the cyclic costructio above, G = C 4r+ (,,..., r) ad its complemet i H, G c = C 4r+ (r +,..., r), are Class 0 by showig that they are D3C. Claim: G, G c are 3-coected. We ote that A is relatively prime to 4r +, ad r A is as well. Sice both A, A cotai multiple elemets, by Lemma 4 the two subgraphs are each 3-coected. Claim: diam(g) = diam(g c ) =. Let v, w V (G) such that vw / E(G). Sice both v ad w are coected to r vertices, but there are oly 4r remaiig distict vertices i G, the v ad w must have a commo eighbor, implyig that there is a two-path betwee them. Sice the same logic follows for G c, diam(g) = diam(g c ) =. We have thus show that whe = 9, 3,..., K ca be edge-partitioed ito two cyclically symmetric D3C graphs, ad hece is Class 0. Through a similar method, we ca exted a similar costructio to graphs of the form K 4r+3, for r =, 3,..., i which the umber of possible eighbors for a give vertex is odd but ot divisible by 4. Theorem 6. Complete graphs of the form K 4r+3 for r =, 3,... ca be edge-partitioed ito two cyclically symmetric D3C subgraphs C 4r+3 (,..., r + ) ad C 4r+ (r + 3,..., r + ), ad hece are -Class 0. Both of these cyclic costructios are for complete graphs with odd umbers of vertices. A ope questio we have is whe it is possible to create cyclically symmetric partitios for K 0, K,...? I Figure 4, we ca use the cyclic costructio for K 9 to demostrate the effect of vertex splittig i creatig a partitio of K 0, showig that it ideed ca be partitioed ito two D3C subgraphs. However it is possible to show that K 0 is ot symmetric -Class 0, ad we as for what eve-vertex complete graphs it is possible. Aother questio is to fid families of costructios for partitioig complete graphs ito > subgraphs which are D3C; we tacle this problem through a differet approach i the ext sectio. 6

7 G G c K Figure 3: -Class 0 costructio for K Figure 4: Splittig v 9 to form -Class 0 K 0 from K 9 5 Existece of -Class 0 graphs for all We will ow prove the existece of a -Class 0 decompositio for each N. I particular, we ivoe the probabilistic method to show that there exists a N such that K ca be edge-partitioed ito cyclically symmetric Class 0 subgraphs. We first show that for each N, there is some N such that K ca be edgepartitioed ito disjoit cyclically symmetric subgraphs, each of which has diameter. To do so, we will prove that for each >, there is a such that each subgraph i a radom decompositio of K ito symmetric subgraphs has diameter. We ca abstract this problem to oe of basic additive umber theory, i which the additio or subtractio of ay two elemets of a set A S represets a two-path to a vertex correspodig to the resultig elemet of S, ad the size of the subsequet sum ad differece set relates to the umber of elemets oe ca reach via a two-path. To clarify this arithmetic, we defie the dihedral sumset as follows. Defiitio. Let N, S = {,..., }, ad A S. The we defie the dihedral sumset 7

8 Partitioig complete graphs o vertices ito =8 symmetric subgraphs Partitioig complete graphs o vertices ito =0 symmetric subgraphs Approx. Probability that first subgraph has diameter Approx. Probability that first subgraph has diameter Number of Vertices () Number of Vertices () Method: We let N be a prime umber, ad fix some N to be the umber of partitios of the complete graph K. Next, we cosider a radom permutatio of,..., ad tae (i order) elemets out at a time, each of which correspods to a radom selectio of a appropriately sized partitio of K such that there will be a total of partitios. We the attempt a simulatio, i which we tae radom selectios of appropriately sized subsets of S for various K, ad evaluate the size of their dihedral sumset. We chose prime graph sizes i order to deal with the 3-coected coditio; the explaatio of this lies i Lemma 4. I this figure we preset the simulated probability that a give radom symmetric subgraph of a complete graph with a prime umber of vertices (x-axis) will be successfully of diameter, i.e. that the dihedral sumset geerated by the correspodig A S is sufficietly large. Figure 5: Simulated probability that a radomly selected subgraph of a complete graph o a prime umber p of vertices will have diameter of A to be the set A A = {x S : x = a + b or x = b a or x = (a + b), for a, b A, b a} Our defiitio of dihedral arithmetic exhausts the ways that oe ca use a pair of edge legths to reach vertices aroud the graph. This is i compariso to the typical sumset, which uder modulo is defied as A+A = {x S : x = a+b mod, for a, b A} [6]. Propositio 7. C (A) has diameter if ad oly if A A = S =. I fact, C (A) has diameter less tha or equal to if ad oly if A A A = S. }{{} times This is ituitive; each additioal dihedral sumset operatio correspods to legth of a path betwee two elemets. We also are iterested i whe a x S ca be reached via two-path usig elemets of a set A. Defiitio. For a give x S, we cosider pairs of elemets a, b S, a b such that either a + b = x, b a = x, or (a + b) = x. We call the evet that both are icluded i a particular subset A a pair evet (a, b) for x i S. We ca get some ituitio o where this probabilistic argumet comes from looig at Figure 5. I simulatio, the larger the size of the complete graph (ad hece the larger the size of a radom subgraph, for fixed ), the more liely the subgraph will be of diameter. The proof begis below. 8

9 Evet class Miimum umber of disjoit evets { (, x ),..., ( x, x )} x {( x, x ) (, x, x ) } { +,... max x, x } { ( 3 (, x + ), (, x + ),..., x, )} ( ) x Figure 6: Distict idepedet pair evets for x i S. Theorem 8. Let N, S = {,..., }. x S, there are at least f() = 8 + pair evets i S, {(a, b ),..., (a f(), b f() )}, such that a,..., a f(), b,..., b f() are all distict elemets of S. Proof. Fix v 0 K as 0, ad cosider some x S as a target, correspodig to v x. Figure 6 presets the umber of distict idepedet pair evets for x S. We deote the classes of evet pairs by the order of their rows i Figure 6 as,, ad 3. There exists oe more possibility: that x itself is a elemet of A S. Groupig this sigle elemet evet with class, we defie f (, x) = x +, f (, x) = max { x, x }, ad f 3 (, x) = ( x). We ote that f 3 is wealy decreasig i x ad f is wealy icreasig i x, ad hece the itersectio of the two fuctios will yield a lower boud for the umber of idepedet pair evets for x. We defie a fuctio f(, x) which selects the greatest umber of idepedet pairs depedig o x, as below: { f (, x) if x f 0 (, x) = 4 f 3 (, x) if x < 4 This will boud the smallest umber of idepedet pair evets. The miimum of this fuctio o S is achieved whe x + = ( x), which occurs whe x =. 4 The miimum for this fuctio will be equivalet to +, ad thus there exist at least 8 f() = + distict pair evets i S for each x S. 8 Defiitio (Idepedet sortig process). We utilize the followig process to sort elemets of S ito distict partitios: we radomly assig each elemet of S to a partitio of A i with probability that a particular partitio will ed up cotaiig a particular elemet. Hece, from the perspective of the partitio itself each elemet will be placed idepedetly with probability. Theorem 9. For N ad N, let A,..., A be defied accordig to the idepedet sortig process. The lim P (diam(c (A i )) = for all i {,..., }) =. I particular, the probability that at least oe of A,..., A is ot of diameter is bouded above by [ ( ) ] 8 + 9

10 Proof. We approach this usig a probabilistic argumet. Cosider the probability of beig able to reach some x S, give A i S. Sice we are assumig idepedet sortig of elemets of S ito partitios with probability, the probability that a sigle fixed pair evet fails will be bouded above by (. ) Sice we have at least f() = + idepedet evets, we ow that 8 P (all the evets fail for x) [ ( ) ] 8 + By this fact ad otig that there are exactly elemets i S, ad subsequetly [ P (at least oe x S is iaccessible i A i ) ( ) ] 8 + ( at least oe x S will be iaccessible P i some A,..., A ) [ ( ) ] 8 +. Hece, lim P (diam(c (A i )) = for all i {,..., }) = I particular, we have just show that for each N, there is some N such that K ca be edge-partitioed ito disjoit cyclically symmetric subgraphs, each of which has diameter. We ca ow coclude with the followig theorem. Theorem 0. For each N, there exists some N such that K ca be edge-partitioed ito cyclically symmetric D3C subgraphs. That is, there exists some large eough such that K is Class 0. Proof. Theorem 9 gives a probabilistic boud o the diameter coditio. To boud the 3-coectivity coditio, cosider the case of prime,, S = {,..., }, ad A,..., A defied accordig to the radom sortig process. It follows from Lemma 4 that restrictig ourselves to prime, for a partitio of S ito A,..., A accordig to our idepedet sortig process, as log as for each i N, A i has more tha oe elemet, the K is partitioable ito 3-coected subgraphs. Note that accordig to our sortig process, ( ) P ( A i ) = + Moreover, by the same logic as i the previous theorem, ( ( ) P ( A i for at least oe i) = + 0 ) ( )

11 Combiig the result from Theorem 9, ( P A i for at least oe i OR at least oe x S will be iaccessible i at least oe of the subsets ) ( ) + ( There exists a prime 0 large eough such that ( ) + ( ) [ + ) [ + ( ) ] 8 + ( ) ] 8 + ad hece there exists by probabilistic argumet a partitio of K 0 D3C subgraphs. ito symmetric 6 Future steps I Figure 7, for selected we ote the smallest -Class 0 complete graph guarateed by Theorem 0, alog with the smallest we have foud by ispectio. Number of partitios Best by asymptotic boud Best ow ? Figure 7: For fixed umber of partitios, existece result implied by theorem It is evidet that the existece result is ot by ay meas a tight boud. Amog possible future steps, we hope to: (i) Recocile the gap betwee smallest observed -Class 0 graphs ad the oes guarateed by probabilistic argumet. A first step would be to more clealy address the 3-coectivity result. (ii) Shift focus from complete graphs to large radom graphs, ad as the same questios. (iii) Ivestigate how strog products ad vertex splittig affect geeral Class 0 graphs, ad specifically assess the cojecture that vertex splittig preserves Class 0 o graphs with two vertices or more. (iv) Begi to aalyze strategies ad characteristics of the multicolor pebblig game.

12 Refereces [] F. R. Chug, Pebblig i hypercubes, SIAM Joural o Discrete Mathematics (989) [] G. Goraly ad R. Hassi, Multi-color pebble motio o graphs, Algorithmica (008). [3] G. Hurlbert, A survey of graph pebblig, Cogr. Numer 39 (999) [4] R. H. Clare, Tom ad G. Hurlbert, Pebblig i diameter two graphs ad products of paths, J. Graph Th. 5 (997) 9 8. [5] G. H. H. K. Czygriow, Adrej ad T. Trotter, A ote o graph pebblig., Graphs ad Combiatorics 8 (00) 9 5. [6] T. C. Tao ad V. H. Vu, Additive Combiatorics. Cambridge Uiversity Press, 006.