USING TOPOLOGICAL METHODS TO FORCE MAXIMAL COMPLETE BIPARTITE SUBGRAPHS OF KNESER GRAPHS

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1 USING TOPOLOGICAL METHODS TO FORCE MAXIMAL COMPLETE BIPARTITE SUBGRAPHS OF KNESER GRAPHS GWEN SPENCER AND FRANCIS EDWARD SU 1. Itroductio Sata likes to ru a lea ad efficiet toy-makig operatio. He also likes to keep up with the latest math developmets. So aturally, whe Sata was buildig a ew factory a few years back, he desiged a iterestig factory ad method for assigig workers to toys that was based o a recet article he read i the Mothly??. Here were some of Sata s costraits. As it turs out, each toy requires elves, but some toys take loger tha other to make. There is oe statio for the fabricatio of each type of toy. Due to a powerful Elf Workers Uio, elves are allowed to take breaks of idetermiate legth betwee projects, but they geerally do t dawdle for too log because they are ot paid for their break time. Also, each worker was to be iitially traied i makig certai toys i cojuctio with certai groups of elves. After readig??, Sata evisioed buildig we he calls a Keser factory, which would have the propert that (i) that each -subset of elves would be traied to make certai toy, ad (ii) disjoit -subsets of elves are ever assiged to the same toy-makig statio. This would esure that wheever elves came back from their break ad were ready to work, they could be immediately assiged to a toy-makig project at a available statio. The first questio Sata asked was: What was the miimal umber of toy-makig statios that he eeded to have i a Keser factory? Notice that the aswer is simple if the umber of elves is less tha, sice oly oe subset of elves ca be workig at ay give time. Therefore, Sata oly eeds oe toy-makig statio. So, suppose that the umber of elves is +k, where k 0. Sata was itrigued to lear of the Keser cojecture (ad its topological proofs, e.g., [?,?,?]) which aswers this questio! Certaily k + statios are eough, because if the elves ad statios are umbered, the we ca assig the i-th statio to ay -subset of elves whose lowest-umbered elf is i, as log as i k + 1 ad all other -subsets are assiged to the k + -d statio. The Keser cojecture claims that k + is also ecessary. Now that Sata has built such a factory, ballooig distributio costs have prompted Sata to cosider opeig a secod factory at the South Pole. He wishes to split the toy-makig statios by movig half of them to the South Pole, which also ecessitates a divisio of the elves. Retraiig elves is expesive. Sata ow asks: 1

2 GWEN SPENCER AND FRANCIS EDWARD SU Is there a way to move half of the toy-makig statios to the South Pole ad divide the elves so that o retraiig is ecessary, each toy is still made at some locatio, ad each factory is a Keser factory? I this article, we aswer Sata s questio i the affirmative; i fact, Sata ca halve the toy-makig statios i ay maer he chooses. Furthermore, we show these ecessitated divisio of elves is ever too lopsided. The settig above ca be rephrased i terms of a theorem that we prove about Keser colorigs of graphs. We provide two proofs, oe with a topological flavor, ad oe combiatorial. Our first method builds o work of Fa[?] ad rel strogly o Fa s geeralizatio of the Lusterik-Schirelma-Borsuk (LSB) thoerem. The Lusterik-Schirelma-Borsuk Theorem. For ay coverig of a - sphere by + 1 closed sets, oe of the sets must cotai a pair of atipodes. The LSB theorem ejoys may equivalet formulatios: the Borsuk-Ulam theorem i topology, ad a combiatorial versio kow as Tucker s lemma. Our secod method of proof exteds Matoušek s recet combiatorial proof of Keser s cojecture that relies o Tucker s lemma; we adapt a costructive proof of a geeralizatio of Tucker s lemma that is equivalet to Fa s LSB geeralizatio. As a itermediate, we will briefly reframe our result i more graph theoretic terms as per Simoyi ad Tardos i [?].. Backgroud Keser graphs are a class of graphs which describe the disjoitess of -subsets of a larger set. We ca thik of Sata s Keser factory as a colorig problem o a Keser graph. Recall the defiitio of a Keser graph. For s r atural umbers, the Keser is the set of s-subsets of {1,,...r}. Two vertices are joied by a edge whe they correspod to disjoit subsets. Figure?? shows a famous example of a Keser graph kow as the Peterse graph (K () 5 ). graph K (s) r is costructed as follows. The vertex set of K (s) r Figure 1. The Peterse Graph: is the Keser graph K () 5. Each vertex i the Peterse graph is a set of two elemets of the set {1,,3,4,5}. A pair of vertices is adjacet i the graph if the sets which correspod to them are disjoit. O the right is a proper colorig of the Peterse graph by the miimal umber of colors: k + = 3. I 1955, Keser cojectured that the chromatic umber of K r (s) is r s + whe r s ad 1 otherwise. 198, Fa proved:

3 FAN S LSB 3 Theorem 1. Let E be a groud set of + k poits, ad suppose each -subset of E is assiged oe of m colors {1,..., m} such that o two disjoit -subsets have the same color. The there exist colors i 1 < i <... < i k+ ad correspodig -subsets A 1,..., A k+ colored i 1,..., i k+ respectively such that j odd A j is disjoit from j eve A j. Notice that this implies Keser s cojecture, because m k+ for the coclusio of the theorem to hold. We prove more i the case m = k + : Theorem. Let E be a groud set of + k poits, ad suppose each -subset of E is assiged oe of k + colors i C = {1,..., k + } such that o two disjoit -subsets have the same color. (a) The for ay divisio of the colors C ito disjoit C 1, C (which differ i size by at most 1 color) there exists a partitio of E ito disjoit sets E 1, E such that all -subsets i E i are colored by colors i C i, ad all colors i C i are exhibited by some -subset i E i. (b) Moreover, each of the pieces E 1, E of the partitio have size greater tha 1/4 the size E. Note that we are halvig the color set C ito sets of size k+ k+ ad. Figure. Colorig k-sets of a ( + k)-set where = k =. Vertices i this graph are poits i the groud set of ( + k) poits, ad edges i the graph are -sets. As a example, cosider the case where = k =, which is show i Figure??. I this case, we are colorig -sets of a 6-elemet set, i.e., the edges of a complete graph o 6 vertices. The leftmost graph shows a Keser-colorig of the edges The other graphs show partitios (which obey the property specified by the Subcolorig Theorem) of the groud set of poits for each possible divisio of the color classes: for istace, the rightmost graph is a partitio that results from partitioig the colors ito the groups blue/yellow ad red/gree. The -subsets which are ot etirely cotaied i a sigle part of the partitio are show as dashed lies; oly the complete -subsets are colored. We will prove the first part of this theorem by slightly extedig Fa s proof of Theorem?? ad observig a few additioal items. 3. Our first proof I order to prove of Theorem?? we will establish a versio of Fa s geeralizatio of the LSB theorem rely o a theorem of Gale cocerig the distributio of poits o a sphere.

4 4 GWEN SPENCER AND FRANCIS EDWARD SU Fa s LSB Geeralizatio. (For Ope Sets) Let k, m be two arbitrary positive itegers. If m ope subsets F 1, F,..., F m of the k-sphere S k cover S k ad if o oe of them cotais a pair of atipodal poits, the there exist k + idices a 1, a,, a k+ such that 1 a 1 < a <... < a k+ m ad F a1 F a F a3 ( 1) k+1 F ak+ where F i deotes the set atipodal to F i. Fa s LSB geeralizatio is geerally stated for F i closed sets [?] but it is easy to establish this versio for ope sets by a stadard ormality argumet. Note that whe m = k + 1 we obtai the LSB theorem, sice the assertio of the theorem caot be true (there simply are ot eough sets to fid such a collectio of a i ) ad therefore at least oe of the F i must cotai a pair of atipodes. The followig result of Gale?? was a crucial igrediet i Baray s proof of Keser s cojecture. Lemma 1 (Gale). There is a distributio of + k poits o S k such that every ope hemisphere of S k cotais at least poits. We are ow ready to proceed with the proof of the mai theorem. We will provide a geeral costructio ad the atted to the two types of cases. Proof. (of Part a of Theorem??): Embed + k poits o S k such that the Gale property is met. Color the -subsets with k + colors (idexed 1,,...,, k + ) such that disjoit -subsets are differetly colored. Let H(x) deote the ope hemisphere cetered at x. Let F i be the set of all poits, x, of S k for which H(x) cotais a -subset of class i. By costructio, the F i are all ope, ad by Lemma??, the F i cover S k. Clearly, F i caot cotai a pair of atipodes sice this would imply the existece of two disjoit -subsets (each cotaied i a ope hemisphere, the two of which are disjoit) of the same class. Sice there are exactly k + colors, the set of idices give i Lemma?? correspods to the set of all colors such that the expressio for the o-empty itersectio simplify to the followig: F 1 F F 3 ( 1) k+1 F k+. That is, there is a otrivial ope itersectio i which every poit has the property that its ope hemisphere cotais -subsets of colors 1, 3, 5, etc, ad its atipode s ope hemisphere cotais -subsets of colors, 4, 6, etc. Sice the set of poits with this property is ope ad there are oly fiitely may groud set poits o the sphere, there is at least oe coordiate poit i the ope itersectio which has the property that the boudary of its ope hemisphere does ot cotai ay groud set poits. Call this poit x. Let all poits i H(x) be i the first set E 1 of the partitio of the origial poit set. Let all poits i H( x) be i the secod set E of the partitio. It is obvious from our costructio that E 1 ad E are disjoit ad that their uio cotais all of the + k groud set poits. As described above, E 1 cotais -subsets of classes 1, 3, 5, etc, where every color is exhibited. Similarly E cotais -subsets of classes, 4, 6, etc, where every colors is exhibited. Sice E 1 H(x) ad E H( x) ay color that is observed o a complete set i E 1 caot be observed i a complete set i E (if this happeed it would have ecessarily followed from a violatio of our origial

5 FAN S LSB 5 assumptio that disjoit -subsets are differetly colored). Thus, all complete subsets of odd-idexed colors are observed i E 1 ad every odd-idexed color is observed i E 1. The aalog for P ad eve classes follows by the same argumet. Sice the labellig of the colors was arbitrary, ay relabellig is equally valid. Thus, ay k+ -subset of colors may be chose to be those which are idexed for membership i E 1 (this is simple: give ay color that is desired to be represeted i the first mai set of the partitio a odd idex). Though?? asserts that we ca fid a partitio of the set of +k poits with the properties described, it tells us othig about the sizes of the pieces of the asserted partitio. To prove Part b of the theorem we costruct bouds o the sizes of these pieces. Deote the larger piece of the partitio by E l ad the smaller piece by E s where E s E l. We will say that a color j is exhibited by E i if some -subset of E i was colored the jth color uder the origial Keser colorig. Proof. (of Part b of Theorem??): First we cosider a obvious boud. Sice E s must have at least k+ colors exhibited, clearly it must have at least that umber of -subsets cotaied withi it: (1) k + ( ) Es. Next, we create a upper boud for E l. First, observe that sice the full set of -subsets is Keser-colored (that is, disjoit sets are colored differetly) the colorigs iduced o the partitio pieces must also be Keser-colorigs. From Part (a) of Theorem??, the -subsets of E l are colored with at most k+ colors. Sice we already observed that E l is Keser-colored, we ca ow apply Keser s cojecture i a reverse style: k+ colors are sufficiet to Keser-color the - subsets of a set that has at most + ( k+ ) poits. If a set has more tha this may poits, the k+ colors will ot be eough to color its -subsets i a Keser way. Sice E l is Keser-colored with at most k+ colors, it must be that E l + ( k+ ). Reiterpretig this as a boud o E s we fid that () E s k + 1. Now suppose that both bouds (??) ad (??) are less tha or equal to 1 4 of the groud set of + k poits. The (??) becomes of size (3) k ( + k), 4 which, whether k is eve or odd, shows that k is certaily less tha or equal to. What does the other boud (??) yield? By assumptio, E s 1 4 (+k) = + k 4. Sice E s must be a iteger, we ca write E s + k 4. We oly make the right side of (??) greater by substitutig + k 4 for E s as follows: (4) k ( k 4 ).

6 6 GWEN SPENCER AND FRANCIS EDWARD SU Now we will substitute i our expressio for k from (??) ito the right side of (??). Sice k : k ( ) ( 4 1 = ) ( ) 1 =. But ( ) 1 equals 0 ad the left side of the iequality is clearly positive, a cotradictio. Hece E s caot be less tha or equal to 1 4 ( + k). 4. Relatio to previous work Now we will discuss the result i the graph theoretic terms used by Simoyi ad Tardos i [?]. For Keser graphs,?? asserts the existece of special complete bipartite subgraphs which have the complete bipartite subgraphs asserted by the Zig-Zag theorem (CITE) as subgraphs. The graphs asserted by?? are special i that they are colored i a particular way, they are i some sese maximal, ad they have the property that the graph iduced by each piece of their bipartitios are also Keser graphs. We say that a complete bipartite subgraph K of a graph G is maximal whe o vertices of G that are ot i K ca be added to K i order to obtai a larger complete bipartite subgraph. A complete bipartite subgraph of a Keser graph is thus maximal whe all vertices ot i K correspod to subsets of the groud set of poits which appear i the uio of the groud poits that correspod to vertices i K. That is, whe the uio of the groud poits correspodig to the vertices of K is the etire origial groud set. Sayig that a bipartite subgraph K of a Keser graph is complete is equivalet to sayig that the groud set of poits correspodig to ay vertex i the first piece of the bipartitio of K is disjoit from the groud sets of poits correspodig to ay vertex i the secod piece of the bipartitio i K ad vice-versa. Suppose that c is a colorig of a Keser graph G by χ(g) colors. I the limitig case the subcolorig theorem states that for ay set W of χ(g) of the colors i c, there exists a maximal complete bipartite subgraph K of G which has for the pieces of the bipartitio of K two sets of vertices: oe set iduces a Keser subgraph colored by the colors of W (with each color i W appearig), the other set iduces a Keser subgraph colored by all the colors ot i W (with all of these colors appearig). 5. A combiatorial Proof Fa provides a alterate formulatio of?? that is a extesio of the combiatorial result kow as Tucker s Lemma. Matoušek recetly published the first combiatorial proof of Keser s cojecture[?] which relies directly o Tucker s Lemma. I this sectio we adapt that proof to fid a origial combiatorial proof that establishes both?? ad Part (a) of??. First we will review Tucker s lemma. Recall that the octahedral subdivisio of the -sphere is the divisio of the sphere iduced by its itersectio with the coordiate hyperplaes i R +1. Also, a barycetric derived subdivisio is a subdivisio derived by successive applicatio of a fiite umber of barycetric subdivisios. We will use the versio of Tucker s lemma stated i [?]:

7 FAN S LSB 7 Tucker s Lemma. Let K be a barycetric subdivisio of the octahedral subdivisio of the -sphere S. Suppose that each vertex of K is assiged a label from {±1, ±,..., ±} i such a way that labels of atipodal vertices sum to zero. The some pair of adjacet vertices of K have labels that sum to zero. Fa s geeralizatio of Tucker s lemma states the followig[?]: Fa s Tucker Geeralizatio. Let K be a barycetric derived subdivisio of the octahedral subdivisio of the -sphere S. Let m be a fixed positive iteger idepedet of. To each vertex of K, let oe of the m umbers {±1, ±,..., ±m} be assiged i such a way that the followig two coditios hold: labels at atipodal vertices sum to zero ad labels at adjacet vertices do ot sum to zero. The there are a odd umber of -simplices whose labels are of the form {k 0, k 1,..., ( 1) k }, where 1 k 0 < k 1 <... < k m. I particular m + 1. As with the LSB theorem, it is easy to check that the limitig case of Fa s Tucker geeralizatio is Tucker s lemma itself. Matoušek s combiatorial proof of Keser s cojecture begis by costructig a barycetric subdivisio of the octahedral subdivisio of S 1. By preservig iformatio about the iclusio of simplices i the subdivisio, ad assigig labels accordig to two cases (oe of which is based o a attempted Keser colorig by k + 1 classes) such that the coditios of Tucker s lemma are met, Matoušek is able to apply Tucker s lemma to obtai a cotradictio. We will slightly exted Matoušek s costructio so that Fa s Tucker geeralizatio ca be used to obtai a combiatorial proof that establishes both?? ad??. The first part of the costructio that follows is take directly from [?]. Let B +k deote the uit ball i R +k uder the l 1 -orm. Let S +k 1 deote its boudary, ad let K 0 be the atural triagulatio of B iduced by the coordiate hyperplaes (where each -dimesioal simplex correspods uiquely to orthat i R ). Call a triagulatio K of B +k a special triagulatio if it refies K 0 ad is atipodally symmetric about the origi. Proof. We will costruct a special triagulatio K, label it i a way that icorporates a proper Keser colorig ad meets the coditios for applyig Fa s Tucker geeralizatio, ad fially apply this result ad iterpret its assertios i our costructio. We begi by defiig the triagulatio K. Let L 0 be the subcomplex of K 0 cosistig of the simplices lyig o S +k 1. Note that the o-empty simplices of L 0 are i oe-to-oe correspodece with ozero vectors from V = { 1, 0, 1} +k. The iclusio relatio o the simplices of L 0 correspods to the relatio o V, where u v if u i v i for all i = 1,,..., ad where 0 1 ad 0 1. Let L 0 be the first barycetric subdivisio of L 0. Thus, the vertices of L 0 are the ceters of gravity of the simplices of L 0 ad the simplices of L 0 correspod to chais of simplices of L 0 uder iclusio. A simplex of L 0 ca be uiquely idetified with a chai i the set V \{(0,..., 0)} uder. Now we defie K: it cosists of the simplices of L 0 ad the coes with the origi for a apex over such simplices. We have costructed a special triagulatio of B +k as i Fa s Tucker s geeralizatio.

8 8 GWEN SPENCER AND FRANCIS EDWARD SU Let E deote a set of +k elemets. Suppose that c is a proper Keser-colorig of the -subsets of E by m colors. I particular, m must be at least k +. For tactical coveiece we will label the colors, + 1,..., + m 1. We will ow defie a labelig of the vertices of K as i Fa s Tucker geeralizatio. These vertices iclude 0, so they ca be idetified with the vectors of V, ad we wat to defie a labelig λ : V {±1, ±,..., ±m}. We fix some arbitrary liear orderig o [+k] that refies the partial orderig accordig to size (which has that A < B implies A < B). Let v V ad defie λ(v) as follows. Cosider the ordered pair (A,B) of disjoit subsets of [ + k] defied by A = {i [ + k] : v i = 1}, B = {i [ + k] : v i = 1}. We distiguish two cases. If A + B (Case 1) the { A + B + 1 if A > B (5) λ(v) = ( A + B + 1) if A < B If A + B 1 (Case ) the at least oe of A ad B has size at least. If, say, A k we defie c(a) as c(a ), where A cosists of the first k elemets of A, ad for B k, c(b) is defied similarly. We set { c(a) if A > B (6) λ(v) = c(b) if A < B Thus, i Case 1 we assig labels i {±1, ±,..., ±( 1)} while labels assiged i Case are i {±, ± + 1,..., ±( + m 1)}. We ow will verify that λ meets the coditios ecessary to apply Fa s Tucker geeralizatio. First, we ote that λ is a well-defied mappig from V to {±1, ±,..., ±(+ m 1)}. To see that λ labels atipodes so that their sum is 0, i.e. that λ(v) = λ( v), we observe that from our defiitios of A ad B, A i = B i where A i deotes the set A that correspods to the ith vector v. Thus labels assiged by both cases will label atipodes with additive iverses. Next, we eed to check that there are o 1-simplices whose vertices labels sum to 0 (that is, there are o complemetary edges). Because of the way we defied λ, ay complemetary edge would have to have had both of its vertices labelled by either Case 1 or Case. Suppose that there is a complemetary edge betwee vertex i ad vertex j. If both labels were assiged by Case 1, the because of our observatio about simplices correspodig to chais i V uder, we would get (after a possible relabellig) that A i A j ad B i B j with at least oe of these iclusios beig proper. But this gives that A i + B i A j + B j so that there is o way that Case 1 could have assiged complemetary labels to the ith ad jth vertices. Suppose both labels were assiged by Case, ad that, without loss of geerality, A i A j ad B i B j. This would mea that the label of the ith vector (which correspods to the color of a k-subset of A i after a possible relabellig) was the egative of the label of the jth vector (which correspods to a k-subset of the same color i B j ). But sice A i A j ad A j B j = this would imply that we had foud two disjoit k-subsets of the same color. Sice c is proper Keser-colorig, this caot be the case. Thus, λ has o complemetray edges. Sice λ has that λ(v) = λ( v) ad cotais o complemetary edges, we ca apply Fa s Tucker geeralizatio. Fa s Tucker geeralizatio gives that there

9 FAN S LSB 9 are a odd umber of + k-simplices i K whose labels are of the form S = {l 0, l 1,..., ( 1) l +k+1 }, where 1 l 0 < l 1 <... < l +k+1 + m 1. I particular there is at least oe + k-simplex with this property. Referrig to our costructio of λ, at least the +k+1 ( 1) = k+ highest of these labels were assiged by Case. Idex these k + vertices that were labelled by Case with the idices {1,,..., k +}. Recallig that the vertices of our +k-simplex correspod to etries of a chai i V uder we fid that (after a possible reidexig): A 1 A... A k+ B 1 B... B k+, where A i B i =. Now let P s = A k+ ad P l = {E \ A k+ }. We make several observatios: Note: a positive label was assiged whe A > B ad a egative labels was assiged whe B > A. Thus, the label j occurrig o the asserted + k simplex i λ would follow from -subset of color i beig cotaied i sets A i... A k+ P s for some i. Similarly, the label r occurrig o the asserted + k simplex i λ would follow from a -subset of color r beig cotaied i the sets B j... B k+ P l for some j. Oe of A k+ or B k+ cotais k-subsets of k+ colors. The other cotais k-subsets of the other k+ colors. Thus, oe of P l or P s cotais k-subsets of at least k+ k+ colors. The other cotais k-subsets of at least other colors. Sice c was a proper Keser-colorig ad P s ad P l are disjoit by costructio, there are o colors that are exhibited by -subsets i both P s ad P l. From our costructio, Tucker s Lemma asserts the existece of a + k simplex with a sequece S of k+ labels (where each label represets a color) whose absolute values icrease mootoically ad whose sigs alterate. From our first observatio, all positive colors i S must have bee exhibited i some A i ad all egative colors i S must have bee exhibited i some B j. Thus A k+ cotais -subsets of each of the positive colors ad B k+ cotais -subsets of each of the egative colors, ad sice A k+ ad B k+ are disjoit by costructio, we have Fa s result from??. Suppose m = k+. All of the -subsets of A k+ are idexed by either exclusively the eve-idexed colors or exclusively the odd-idexed colors. Similarly, all of the -subsets of B k+ are idexed by the other parity colors. Sice there are o colors exhibited by P s ad P l which are ot exhibited by A k+ ad B k+ respectively, the same property holds for these sets. Thus, P s ad P l are the two pieces of the partitio described i??. Sice the idices assiged to the colors were arbitrary, this result holds for ay relabellig: for ay set of k+ colors, there exists a partitio which meets the coditios described i?? i which they appear i the same piece of the partitio. Thus, we coclude the proof. Departmet of Mathematics, Harvey Mudd College, Claremot, CA address: gspecer@hmc.edu, su@math.hmc.edu