In Search of the Fractional Four Color Theorem. Ari Nieh Gregory Levin, Advisor

Transcription

1 In Serch of the Frctionl Four Color Theorem by Ari Nieh Gregory Levin, Advisor Advisor: Second Reder: (Arthur Benjmin) My 2001 Deprtment of Mthemtics

2 Abstrct In Serch of the Frctionl Four Color Theorem by Ari Nieh My 2001 Tit showed in 1878 tht the Four Color Theorem is equivlent to being ble to three-color the edges of ny plnr, three-regulr, two-edge-connected grph. Not surprisingly, this equivlent problem proved to be eqully difficult. We consider the problem of frctionl colorings, which resemble ordinry colorings but llow for some degree of cheting. Hppily, it is known tht every plnr three-regulr, two-edge-connected grph is frctionlly three-edge-colorble. Is there n nlogue to Tit s Theorem which would llow us to derive the Frctionl Four Color Theorem from this edge-coloring result?

3 Tble of Contents List of Figures iii Chpter 1: Introduction 1 Chpter 2: Bckground nd Definitions Grph Theory Essentils Proof of Tit s Theorem Frctionl Grph Theory Chpter 3: Edge Cuts nd Primitive Grphs Color Prity in Edge Cuts Exmples of Primitive Grphs Chpter 4: Tit s Theorem Revisited Another Interprettion of Tit s Theorem Generliztion to b = Chpter 5: Conclusion 21 Appix A: Appix 22 A.1 The Subdditivity Lemm A.2 Liner Progrmming A.3 Mtlb Code

4 Bibliogrphy 33 ii

5 List of Figures 2.1 Proof of Tit s Theorem A 2-fold Coloring of the 5-cycle Our Grph Before nd After Cutting nd Rejoining Edges Primitive Coloring of Petersen s Grph Primitive Coloring of Dodechedron Tit s Theorem Another View of Tit s Theorem Conditions for Consistency of f Continuous Deformtions of Pth in the Plne iii

6 Acknowledgments I would like to thnk my dvisor, Professor Greg Levin, for his invluble guidnce in writing this thesis. I would lso like to thnk Professor Lesley Wrd nd Professor Art Benjmin for their feedbck. iv

7 Chpter 1 Introduction The Four Color Theorem, which confounded grph theorists for more thn century, lso implies the result of interest herein, nmely, the Frctionl Four Color Theorem. However, the only known proofs of the Four Color Theorem re extremely complex nd require hevy use of computers. Does simple proof of the Frctionl Four Color Theorem exist? We investigte this possibility by serching for frctionl nlogue to Tit s Theorem. In Chpter 2, we will give bckground nd definitions. Chpter 3 will consist of progress nd results regrding edge cuts in relevnt plnr grphs, nd Chpter 4 will focus on our chrcteriztion of primitive grphs. Finlly, Chpter 5 will discuss our ttempted generliztion of Tit s Theorem. The Four Color Problem In 1852, while coloring mp of the counties of Englnd, Frncis Guthrie found tht four colors were sufficient to ensure tht djcent counties were ssigned different colors. Nturlly, he wondered if it ws necessrily true for ll mps. In grph theory terms, this corresponds to coloring the fces of plnr grph such tht no two fces tht shre n edge hve the sme color. Becuse the vertices of the plnr dul of grph correspond to the fces of the originl grph, this problem is equivlent to showing tht the chromtic number of plnr grph never exceeds four.

8 2 Grph theorists puzzled over this problem for yers. Severl, including Kempe nd Tit, cme up with fulty proofs. (In fct, the problem hs remined fvorite of crnks to this dy.) It ws finlly proven by Appel nd Hken in 1976 through extensive use of computers. Unfortuntely, mny found the method of proof somewht unstisfying. Even tody, lthough their pproch hs been simplified, no simple proof is known to exist. Tit s Theorem In 1873, Tit proved tht coloring the edges of ny 3-regulr, 2-edge-connected plnr grph with three colors such tht incident edges did not shre the sme color ws equivlent to properly four-coloring its fces. This implied tht the Four Color Problem could be solved by finding wy to properly three-color the edges of ll such grphs. Unfortuntely, this problem proved no more trctble thn the originl. Frctionl Grph Theory Frctionl grph theory is field of grph theory tht defines rtionl-vlued equivlents of normlly integer-vlued grph theory concepts. For exmple, the chromtic number of grph, typiclly denoted χ(g), which represents the fewest number of colors necessry to color the vertices of grph such tht no two djcent vertices re the sme color, is necessrily n integer. The frctionl chromtic number χ f (G) denotes similr concept, but cn tke on rtionl vlues. Our gol in this pper is proof of the Frctionl Four Color Theorem, nmely, tht every plnr grph hs χ f (G) 4. Becuse it is lwys the cse tht χ f χ, the Frctionl Four Color Theorem is implied by the norml Four Color Theorem. However, to find n lternte wy of getting t the Frctionl Four Color Theorem, we might try to ppel to the fct tht the frctionl edge chromtic number

9 3 of ny 3-regulr, 2-edge-connected plnr grph is known to be three. If there existed frctionl nlogue of Tit s Theorem, which would (in n idel world) tke 3-edge-coloring to 4-fce-coloring in some plesntly frctionl wy, then the truth of the Frctionl Four Color Theorem could be verified without ppeling to the originl, non-frctionl problem.

10 Chpter 2 Bckground nd Definitions This chpter contins the mthemticl bckground nd terminology necessry for the reminder of the pper. In Section 2.1 we review the bsics of grph theory. In Section 2.2 we prove the relevnt direction of Tit s Theorem. Section 2.3 introduces frctionl grph theory nd defines the specific nottion nd terms used in our reserch. 2.1 Grph Theory Essentils A grph G consists of two sets- vertex set V nd n edge set E consisting of size 2 subsets of V. 1 Grphs re commonly represented visully, with vertices drwn s points, nd edges s lines or curves between their two vertices. A subgrph G of G is grph with vertex set V nd edge set E such tht V V nd E E. A pth is sequence of distinct djcent vertices. A grph is connected if there exists pth between ny two vertices. A component of G is mximl connected subgrph. We denote the set of edges between two disjoint sets of vertices S nd T by [S, T ]. Such set is n edge cut if S nd T re nonempty nd S T = V. Note tht deleting n edge cut necessrily disconnects connected grph. An edge cut of size one is clled cut-edge. A grph is k-edge-connected if there does not exist n edge cut of size less thn k. A grph is clled plnr if it cn be drwn in the plne without edge-crossings. 1 This definition, which suffices for this pper, is ctully the definition of simple grph. A simple grph does not contin loops (edges from vertex to itself) or multiple edges between two vertices.

11 5 Such drwing is clled crossing-free plnr embedding. A grph thus embedded is plne grph. A plnr embedding determines fces of grph, which re the regions bounded by its edges. Assigning color to ech vertex of grph is clled vertex coloring. A coloring is clled proper if no two djcent vertices receive the sme color. We will frequently denote colors with numbers for the ske of clrity. The chromtic number χ(g) of grph is the smllest number of colors with which the vertices of G my be properly colored. Edge colorings, proper edge colorings nd edge chromtic number re defined nlogously. Becuse only proper colorings re of interest in this pper, we will henceforth buse our terminology by omission of the qulifier proper. The number of edges incident to vertex is the degree of the vertex. A grph is clled k-regulr if every vertex hs degree k. A 1-regulr grph (or subgrph) is clled perfect mtching. A cycle is connected grph (or subgrph) in which ech vertex hs degree two. It is simplest to think of cycle s closed, non-intersecting loop of vertices nd edges. By definition, it is cler tht tht ny 2-regulr grph is the union of disjoint cycles. 2.2 Proof of Tit s Theorem Theorem 1 (Tit 1878) A 2-edge-connected 3-regulr plne grph embedding is 4-fcecolorble if nd only if it is 3-edge-colorble. We will show only the bckwrd direction, s the other direction is irrelevnt to our reserch. Proof: We re given 3-regulr 2-edge-connected plne grph G. Assume tht the edges hve been properly colored with colors, b, nd c. Let E, E b, nd E c be

12 6 the subgrphs formed by edges of their respective colors. Let H 0 = E E b nd H 1 = E E c. Both H 0 nd H 1 re 2-regulr, becuse they re the originl grph with one color deleted, which removes one edge from ech vertex. Therefore, both H 0 nd H 1 re unions of disjoint cycles. As such, ech cn be used to ssign binry strings of length two to the fces of G s shown. G H H 0 1 G c b b c c b b c c b 1_ b b 0 1 _0 c c b c c b Figure 2.1: Proof of Tit s Theorem To determine the first bit of ech fce string, exmine H 0. Assign the first bit vlue one if it is contined in n odd number of cycles, nd zero otherwise. (In our exmple, there re no nested cycles- so the number of cycles region is contined in is lwys zero or one.) Similrly, using the subgrph H 1, let the second bit be one if it is contined in n odd number of cycles, nd zero otherwise. Then, use the two bits ssigned to ech fce to construct 4-fce-coloring with colors 00, 01, 10, nd 11. This coloring is proper becuse ech edge ppers in t lest one of H 0 nd H 1, so djcent fces must differ in t lest one of the two bits. 2.3 Frctionl Grph Theory In this section, we will define terms specific to frctionl grph theory. A b-fold vertex coloring of grph G ssigns to ech vertex set of b distinct colors. Such coloring is proper if djcent vertices receive disjoint color sets.

13 7 The b-fold chromtic number χ b (G) of grph is the minimum number of colors necessry to properly b-fold color the vertices of grph Figure 2.2: A 2-fold Coloring of the 5-cycle Notice tht χ b (G) b χ(g) for ll b Z, becuse simply replicting n ordinry coloring will yield b-fold coloring of b χ(g) colors. The frctionl chromtic number is defined by χ f (G) = lim b χ b (G) b (2.1) This limit is gurnteed to exist by the subdditivity lemm, nd is gurnteed to be chieved for some vlue of b due to results from liner progrmming 2. It is lso necessrily chieved for every integer multiple of the smllest such b. The frctionl quntities for edge nd fce colorings re defined in n nlogous mnner. In trying to prove the Frctionl Four Color Theorem, our gol is to find some wy of trnsforming b-fold 3b-edge-coloring into t-fold 4t-fce-coloring. Becuse Tit s theorem gurntees this for b = 1, we wish to consider b-fold 3b-edgecolorings tht do not directly contin ordinry (non-frctionl) 3-edge-colorings. For this reson, it is necessry to define clss of b-fold colorings which re fundmentlly mny-fold, nd not merely extensions of ordinry colorings. 2 See ppix for more detils

14 8 We cll b-fold 3b-edge-coloring of 3-regulr grph primitive if every possible pir of colors ppers on t lest one edge. (Note tht in such coloring, ech of the 3b colors ppers on the edges round ny given vertex exctly once; tht is, ll the edges contining ny prticulr color re perfect mtching.) Any non-primitive b-fold 3b-edge-coloring must contin n ordinry 3-edge-coloring in the following sense given two colors nd b which never pper on the sme edge, color the remining edges with c. Becuse both nd b re incident to ech vertex exctly once nd on different edges, the remining edges must form perfect mtching. Therefore, the grph is properly 3-edge-colored with, b, nd c, nd our frctionl coloring contins n ordinry coloring. This definition chrcterizes the colorings which will be useful for our nlogue. When we exmine non-primitive coloring, Tit s Theorem yields n obvious nd somewht unvoidble ordinry 4-fce-coloring, preventing us from determining wht kind of t-fold 4t-fce-coloring should be implied by our hypotheticl nlogue. Cll 3-regulr, 2-edge-connected plnr grph strongly primitive if it hs no non-primitive b-fold 3b-edge-coloring for ny positive integer b nd hence no ordinry 3-edge-coloring (such grph would be counterexmple to the Four Color Theorem, but it is convenient to define it nevertheless).

15 Chpter 3 Edge Cuts nd Primitive Grphs We were interested in exmples to guide construction of smllest counterexmples of the Frctionl Four Color Theorem. This motivted our study of properties of primitive grphs. To investigte these properties, we derived results regrding edge cuts in b-fold 3b-edge-colored grphs. Define 3b-grph s b-fold 3b-edgecolored, 2-edge-connected, 3-regulr grph. The results in this chpter re not directly relted to our conclusions, nd re included for completeness. 3.1 Color Prity in Edge Cuts Theorem 2 In ny edge cut of 3b-grph, ech color ppers n equl number of times modulo 2. Proof: Let S V (G) define n edge cut M = [S, S c ], nd begin with S = V (G) nd S c = {φ}. Notice tht initilly, the prities of ll colors ppering in M re vcuously equl. One t time, move vertices from S to S c until M is the edge cut in question. Becuse ech of the 3b colors is incident to ny vertex exctly once, ech move either dds or subtrcts one from the number of ppernces of ech color in M, which leves their reltive prities unchnged. Therefore, the edge cut must use every color with the sme prity. Corollry: In ny edge cut of size two in such grph, both edges must be identiclly colored. Proof: Ech edge only uses b colors. Therefore, not every one of the 3b colors

16 10 cn be used by two edges. Since t lest one color is used zero times, ll colors must be used n even number of times. Therefore, no color cn pper in one edge s color set nd not in the other s, nd the two color sets must be identicl. We cn now prove tht the smllest counterexmple to the Four Color Theorem is 3-edge-connected. (Unfortuntely, Tit hd gotten to this century before we did using reltively simple methods of norml grph theory.) Corollry: The smllest strongly primitive 3-regulr 2-edge-connected plnr grph must be 3-edge-connected. Proof: Assume tht there exists n edge cut of size two. Then the two edges re identiclly colored, nd two smller grphs my be formed by cutting both edges nd joining them internlly. Figure 3.1: Our Grph Before nd After Cutting nd Rejoining Edges Since the two smller grphs formed cnnot be strongly primitive, 3-color their edges. Then, fter synchronizing the two colorings by pproprite permuttions, cut nd rejoin the edges to form the originl grph, colored in non-primitive wy. This is contrdiction, so no such edge cut cn exist. There re severl nerly identicl corollries using similr rguments.

17 11 Corollry: The smllest strongly primitive 3-regulr 2-edge-connected grph must be 3-edge-connected. Corollry: The smllest primitive 3-regulr 2-edge-connected plnr grph must be 3-edge-connected. Corollry: The smllest primitive 3-regulr 2-edge-connected grph must be 3-edge-connected. Finlly, since we hve eliminted the possibility of size two edge cuts, we exmine the next cse. Corollry: Given ny edge cut of size three in such grph, either ech color is represented once, or hlf of the colors re represented twice nd ech edge shres distinct hlf of its color set with ech of the other two edges. If b = 2, then only the former cse is possible. The proof is similr to tht of Theorem 3.1, beginning with M s the given edge cut, nd showing invrince of the prities of certin color combintions to eliminte the ltter cse. 3.2 Exmples of Primitive Grphs In n effort to find ny exmple of primitive 3b-grph, plnr or not, we set b = 2 nd looked for grphs tht used every possible pir of colors on n edge. To find the smllest possible exmple, we looked t the 15 possible pirs of two colors. Theorem 3 Petersen s grph is the smllest primitive 3b-grph for b = 2. Proof: By our definition of primitivity, ech possible color pir must pper together on some edge. Since there re ( 6 2) = 15 such pirs, ny primitive grph for b = 2 must hve t lest 15 edges. Becuse 3b-grphs re 3-regulr, this is equivlent to hving t lest 10 vertices. Petersen s grph stisfies these lower limits nd is therefore minimum. Uniqueness follows by considering cses.

18 Figure 3.2: Primitive Coloring of Petersen s Grph Unfortuntely, Petersen s grph is not plnr, so it is not useful exmple for exmining the reltionship between the b-fold edge-coloring nd ny corresponding t-fold fce-coloring. After some muttion of Petersen s grph, we found tht the dodechedron is plnr 3b-grph with primitive 2-fold 6-edge-coloring.

19 Figure 3.3: Primitive Coloring of Dodechedron

20 Chpter 4 Tit s Theorem Revisited This chpter exmines more generl interprettion of Tit s Theorem. In Section 4.1 we derive the lternte interprettion of Tit s Theorem. In Section 4.2 we generlize the logic behind the theorem. We then ttempt to pply it to the cse b = 2. All grphs in this chpter re ssumed to be plnr unless otherwise specified. 4.1 Another Interprettion of Tit s Theorem In our proof of Tit s Theorem, note tht color lwys ppers on edges between fces whose binry strings differ in both bits. This is becuse the edges pper in both H 0 nd H 1. Similrly, edges with color b lwys pper between fces whose strings differ in only the first bit, becuse b is only in H 0. Lstly, edges with color c must pper between fces whose strings differ in only the second bit, becuse c is only in H 1. G H H 0 1 G c b b c c b b c c b 1_ b b 0 1 _0 c c b c c b Figure 4.1: Tit s Theorem

21 15 In other words, the fce strings generted by Tit s Theorem determine unique function f from the edge color sets to binry edge strings. In this cse, f() = 11, f(b) = 10, nd f(c) = 11. In fct, this function determines the binry fce strings generted by Tit s Theorem in the following mnner: use f to ssign binry string to ech edge. Assign the unbounded fce the zero string. Then, use the opertion of binry XOR to fill in fce strings. In other words, when stepping over n edge between fces F 1 nd F 2 seprted by edge color set C, give the string on F 1 vlue equl to the string on F 2 dded bitwise modulo 2 to f(c). c b b b c c Figure 4.2: Another View of Tit s Theorem While it is cler tht vlid f results from the 4-fce-coloring generted by Tit s theorem, we cn mke more generl clim bout when function f from ll possible edge color sets into ll binry strings of certin length cn be used to generte t-fold 4t-fce-coloring. (Nturlly, becuse we re ttempting to find n nlogue to Tit s Theorem, we would rther strt with 3b-edge-coloring nd devise n f which will generte fce strings in this mnner.) We cll n ssignment of binry fce strings proper if djcent fces receive different strings. We denote this ssignment h f (F ), function mpping fces to binry strings s defined by the binry XORs generted by f. It is cler the h f is proper if nd only if it mps djcent fces to different strings. Theorem 4 Given function f tht mps size b subsets of our 3b edge colors to binry

22 16 strings of length l, h f is proper on ny plnr 3b-grph if nd only if the following two conditions hold: (I) The zero string is not in the rnge of f. (II) If A, B, nd C re disjoint edge color sets ech of size b, then the binry sum (or XOR) of f(a), f(b), nd f(c) is the zero string. Note tht in the second condition, A B C is the set of ll possible fce colors, nd A, B, nd C re sets tht could pper on the three edges incident to single vertex. Proof:It is firly cler tht these conditions re necessry. If there existed color set C such tht f(c) ws the zero string, then ny grph with the color set C on n edge would hve two djcent fces F 1 nd F 2 with identicl binry strings. Similrly, if the second condition did not hold for some disjoint size b color sets A, B, nd, C, ny grph with vertex v round which A, B, nd C ppered could not hve consistent fce strings. If we were to strt t D, one of the three fces djcent to v nd follow closed loop round it pssing through the three edges, we would t D, nd the binry difference between h f (D) nd itself would be nonzero. This is impossible. F 1 B C F 2 D A Figure 4.3: h f (F 1 ) = h f (F 2 ), h f (D) = h f (D) + f(a) + f(b) + f(c) Showing tht these conditions re sufficient is slightly more involved. We must show tht h f is well-defined. Tht is, given fce F of 3b-grph, suppose we

23 17 consider multiple pths in the plne between the unbounded fce nd F. How do we know tht the sum of the binry strings on the edges through which pth psses is constnt for ll pths? To prove this, we use condition (II). Becuse the plne is simply connected, ll such pths cn be continuously deformed into ech other. Therefore, we cn discuss wht hppens when pth is deformed through vertex. Becuse the sum of the strings on the edges surrounding ny vertex is the zero string, the binry sum of the edges through which pth psses is invrint under continuous deformtions of tht pth, s shown. Therefore, h f is well-defined. Finlly, the first condition clerly suffices to ensure tht djcent fce strings re different. A Sum(s) = f(a) s t B C Sum(t) = f(b) + f(c) = f(a) Figure 4.4: f(a) + f(b) + f(c) = Generliztion to b = 2 Now tht we know precisely wht conditions on f generte consistent, proper fce strings on ny plnr 3b-grph, we cn ttempt to pply this to the cse b = 2 nd find n nlogue of Tit s Theorem. If one does exist, proving it for b = 2 will likely shed some light on the generl cse. If one does not exist, then b = 2 might very well be the esiest counterexmple. Our pln of ttck, then, is to find some f stisfying our properties, nd then nother function g mpping fce strings to fce color sets of size t chosen from 4t

24 18 colors. Note tht in our originl proof of Tit s Theorem, g ws somewht trivil, s t = 1. Clerly, the necessry nd sufficient condition on g for it to complete the nlogue re tht it mps potentilly djcent fce strings (tht is, fce strings whose binry difference is in the rnge of f) to disjoint fce color sets. To find possible f, we considered ll 1-bit binry functions which stisfied condition (II) on f. (Clerly, only f = 1 could stisfy condition (I) for strings of length 1.) This set is n belin group under binry XOR (becuse the XOR opertion preserves condition (II)), nd ech of its 32 elements hs order 2, so it is isomorphic to Z 2 Z 2 Z 2 Z 2 Z 2. This set is esiest viewed s mtrix whose rows re the fifteen color pirs nd whose columns re the functions. (Mtlb code for generting this mtrix is included in the ppix.) It is not hrd to see tht ll possible functions f fulfilling the desired conditions cn be mde by smshing together these 1-bit functions. In order to choose n f which will put sufficient structure on the fce strings of our grph to enble us to find vlid g, we must pick some number of columns from our mtrix. Linerly indepent sets of columns re the only ones of interest, becuse linerly depent columns will not provide dditionl structure on the fce strings. Clerly,

25 19 fewer thn three will result in violtion of condition (I). Picking three columns will not suffice, either. If we choose submtrix of three columns with no row of zeroes (which is equivlent to the first condition), ll possible nonzero binry strings of length three pper in the rnge of f. Clerly, there is no wy to choose g such tht fce strings whose difference is in the rnge of f receive disjoint color sets, becuse the rnge of f is ll of {0, 1} 3. Becuse our group is isomorphic to Z 2 Z 2 Z 2 Z 2 Z 2, the column spce of ny subset cnnot hve dimension greter thn five. Therefore, if we seek n f tht leds to working nlogue, we must pick strings of length four or five. We hve shown by computer tht ll sets of four linerly indepent columns hve t lest eleven different rows. Tht is, the rnge of f hs size eleven. We know tht ll possible fce strings in {0, 1} 4 could occur in some grph, becuse our submtrix hs rnk four. Therefore, given fce color string, g() must be disjoint from g(b) for t lest eleven strings b {0, 1} 4. It follows tht for ny fce color string, g() cn overlp t most four other color sets. 1 If g is one-to-one, we cn model g s hypergrph problem. A hypergrph is vertex set V nd hyperedge set E composed of non-empty subsets of V. If g exists in this cse, then there must exist hypergrph with 4t vertices (representing the colors) nd 16 t-hyperedges (tht is, color sets of size t). It must lso be the cse tht none of these 16 hyperedges is djcent to more thn four others. Such hypergrph is combintoril impossibility. Theorem There does not exist hypergrph with 4t vertices nd 16 t-hyperedges such tht no hyperedge is djcent to more thn four others. Proof:Assume tht such hypergrph G exists. Choose hyperedge, nd consider the subgrph G 1 induced by removing ll vertices contined in. G 1 hs 3t vertices nd t lest 11 hyperedges, becuse ws djcent to t most four = 16

26 20 others. Now pick hyperedge b in G 1, nd consider the subgrph G 2 induced by removing ll vertices contined in b. By similr resoning, G 2 hs 2t vertices nd t lest six hyperedges. I clim tht G 2 is its own component in G. Consider hyperedge e in G 2. Becuse e contins t vertices, only one of the other hyperedges in G 2 (nmely, the one contining the other t vertices) cn be disjoint from e. Therefore, e is djcent to four other hyperedges in G 2, so none of the vertices in e cn be in hyperedges from outside of G 2. Becuse e ws selected without loss of generlity, it follows tht no vertex in G 2 cn be prt of n outside edge, so G 2 is its own component. It is lso cler tht G 2 must hve exctly six hyperedges. If it hd more, then e would hve to be djcent to more thn four others. This implies tht the hypergrph G G 2, which is well defined becuse no edges connect G 2 to the rest of G, hs 2t vertices nd ten hyperedges. This is clerly impossible, s we hve just shown tht 2t vertex hypergrph with these conditions cn hve no more thn six hyperedges. By contrdiction, G does not exist. However, we do not know tht g must be one-to-one. In fct, if our edgecoloring is not primitive, g tht is not one-to-one will properly four-color the fces of our grph! This leves us with the question- if we hve primitive 3bedge-coloring, must g be one-to-one? Becuse non-primitive colorings yield obvious solutions vi Tit s Theorem, this would eliminte the cse of length four binry strings for b = 2 nd llow us to focus on length five strings.

27 Chpter 5 Conclusion Our reserch rises severl unnswered questions worthy of further investigtion. -Is Petersen s grph the unique minimum primitive grph, indepent of b? -Does primitivity of 3b-edge-coloring imply tht g must be one-to-one? -Cn n nlogue of Tit s Theorem ctully be found for b = 2, using {0, 1} 5 s our rnge of f nd domin of g? -If so, how cn it be exted to ll vlues of b? -Is there different wy to generlize Tit s Theorem to the frctionl cse?

28 Appix A Appix A.1 The Subdditivity Lemm A function f mpping the positive integers to R is subdditive if f() + f(b) f( + b) for ll, b. The lemm itself sttes tht if f is non-negtive nd subdditive, the limit s n of f(n) n f(n) exists nd is equl to the infimum of for ll n. n A.2 Liner Progrmming An lternte wy to define the chromtic number of grph is through liner progrmming. An indepent set is set of vertices, none of which hve ny edges between them. χ(g) is equl to the solution of the problem: Minimize c T x subject to Ax b where c nd b re ppropritely sized vectors of ll ones, A is the vertex-indepence set djcency mtrix, nd x is vector of zeroes nd ones. We cn view the vector x s picking set of indepent sets such tht every vertex is contined in t lest one. A miniml such cover of the vertices is miniml coloring, where one color is ssigned to the vertices of ech indepent set. Therefore, the solution to this problem is the minimum number of colors needed to properly color the vertices of grph- the chromtic number. However, if we relx the requirement tht entries of x re from {0, 1}, nd insted llow them to be chosen from [0, 1], the solution to our liner progrm is insted χ f (G), the frctionl chromtic number.

29 23 A.3 Mtlb Code The following mtlb code, written by Professor Greg Levin, ws instrumentl in deling with exmples in our reserch. % CHOOSE(n,k) returns "n choose k" function m = choose(n,k) %"out of rnge" input if k>n m = 0; return; %negtive input if (k<0) (n<0) m = 0; return; %non-integrl input if (k = floor(k)) (n = floor(n)) m = 0; return;

30 24 if k > (n/2) k = n-k; m = prod((n-k+1):n)/prod(1:k); % CHOOSE4 Itertes through ll 21 C 4 col sets of G % This routine runs thru ll size four sets of columns of the % mtrix (group) G generted by MAKEG.M, nd in ech corresponding % 15x4 submtrix, counts the number of *distinct* rows counter = zeros(1,15); foo = [ ]; % Check ech size four subset of G s columns, designted M subset = [ones(4,1);zeros(27,1)]; while (subset = Inf) M = G(:,find(subset)); count = distrows(m); % check for zero rows if ll(sum(m,2)) counter(1) = counter(1)+1; % report new row count else if counter(count) == 0 count

31 25 counter(count) = counter(count)+1; if count == 7 foo = [foo ; find(subset) ]; % yeh = M; % return % report every 1000th subset if mod(sum(counter),1000)==0 totl = sum(counter) subset = nextsub(subset); % CHOOSE4 Itertes through ll 21 C 4 col sets of G % This routine runs thru ll size four sets of columns of the % mtrix (group) G generted by MAKEG.M, nd in ech corresponding % 15x4 submtrix, counts the number of *distinct* rows counter = zeros(1,15); % Generte the indictor strings for 31 C 4 (time consuming) C31_4 = mkecomb(31,4); % Check ech size four subset of G s columns, designted M

32 26 for subset=c31_4 M = G(:,find(subset)); count = distrows(m); % report new row count if counter(count) == 0 count counter(count) = counter(count)+1; % report every 1000th subset if mod(sum(counter),1000)==0 totl = sum(counter) if count == 9 yeh = M; return % DISTROWS(M) Counts the distinct rows of the mtrix M function count = distrows(m) rows = size(m,1); count = rows; i=1; while (i<rows)

33 27 for j=(i+1):rows if isequl( M(i,:), M(j,:) ) count = count-1; brek i = i+1; % MAKECOLD(N,K) Returns n rry representing N choose K % MAKECOLD(N,K) returns n N-by-(N choose K) rry contining % every possible length N binry string with K ones. % Note tht MAKECOMB is fster, non-recursive routine % with the sme functionlity. function C = mkecold(n,k) % We run recursively by putting 1s bove mkecomb(n-1,k-1) % nd 0s bove mkecomb(n-1,k). % First check consistency C = Inf; if (fix(n) = n) (fix(k) = k) return if (n < 0 k < 0 n < k)

34 28 return % Next hndle bse cses if (n==1) C = k; return elseif (k==0) C = zeros(n,1); return elseif (k==n) C = ones(n,1); return % Now hndle recursion C1 = [ ones(1,choose(n-1,k-1)) ; mkecomb(n-1,k-1)]; C2 = [zeros(1,choose(n-1,k)) ; mkecomb(n-1,k)]; C = [C1, C2]; % MAKECOMB(N,K) Returns n rry representing N choose K % MAKECOMB(N,K) returns n N-by-(N choose K) rry contining % every possible length N binry string with K ones. % Unlike the recursive MAKECOLD, MAKECOMB uses the % NEXTSUB subroutine.

35 29 function C = mkecomb(n,k) % We run recursively by putting 1s bove mkecomb(n-1,k-1) % nd 0s bove mkecomb(n-1,k). % First check consistency C = Inf; if (fix(n) = n) (fix(k) = k) return if (n < 0 k < 0 n < k) return % initilize nck = choose(n,k); C = zeros(n,nck); X = [ones(k,1);zeros(n-k,1)]; i = 0; % fill in C while X = Inf i = i+1; C(:,i) = X;

36 30 X = nextsub(x); % MAKEG Genertes the funky edge-color group Z_2ˆ5 % This rgumentless function returns the 15x31 mtrix whose % columns re the binry vectors in the group of permissible % bit ssignments to 2-fold six coloring of 3-regulr grph. % It lso cretes G1, G2 nd G3, the submtrices corresponding % to size 1, 2 nd 3 color sets. %function G = mkeg() G = zeros(15,31); C62 = mkecomb(6,2); % Generte the 6 elements (columns) corresponding to the six colors G1 = ones(15,6) - C62 ; % Generte the 6 C 2 elements corresponding to color pirs G2 = mod(g1*c62,2); % Generte the (6 C 3)/2 elements corresponding to color triples C63 = [ones(1,choose(5,2)) ; mkecomb(5,2)]; G3 = mod(g1*c63,2);

37 31 G = [G1,G2,G3]; % NEXTSUB(X) Returns the next eqully sized subset % If X is binry column vector with k ones (representing % size k subset of n n set), the "next" size k subset % of n n set is returned, or Inf if X is the "lst" subset function Y = nextsub(x) % check for column vector if (size(x,2) > 1) Y = thts not col vector, you dolt return % initilize vlues n = size(x,1); elts = find(x); k = size(elts,1); if k==0 Y = Inf; return

38 32 % find which bit is free to move move = k; while elts(move)==(n-(k-move)) move = move-1; if move==0 Y = Inf; return; % move bit forwrd one, nd put ll following bits right fter next = elts(move)+1; for i=move:k elts(i) = next+(i-move); % fill in Y Y = zeros(n,1); Y(elts,:) = ones(k,1);

39 Bibliogrphy [1] Edwrd R. Scheinermn nd Dniel H. Ullmn. Frctionl Grph Theory. John Wiley & Sons, Inc., [2] Dougls B. West. Introduction to Grph Theory. Prentice Hll, 1996.