Admssble Permutatos ad the CP, the AP ad the TSP oward Klema Prof. Emertus, QCC, Cty Uversty of New York
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Table of Cotets Preface page 5 Itroducto page 6 P = NP? -admssble Permutatos ad the amlto Crcut Problem. Itroducto page 6. Theorems page 8. Algorthms G ad D page 4.4 The Probablty of Success page 56.5 Further Results page 6.6 Geeral -admssble Permutatos page 75.7 A eurstc for the Travelg Salesma Problem page 76.8 Notes page 777 Example. page 6 Example. page 7 Example. page 8 Example.4 page 89 Example.5 page 9 Refereces page Admssble permutatos ad a Algorthm of Freze. Itroducto. page. Symbols, Deftos ad Prelmary Theorems page. The Algorthm page 8
Example. page 4 Refereces page 7 The Floyd-Warshall Algorthm, the AP ad the TSP. Itroducto page 9. Theorems page. The Algorthm page 8 Phase I: Admssble Permutatos page 8 Phase II: The Floyd-Warshall Algorthm, Negatve Cycles page 4 ad the AP Phase III: The F-W Algorthm, No-egatve Cycles ad page 44 the TSP Example. page Example. page 4 Example. page 6 Example.4 page 4 Example.5 page 46 Example.6 page 64 Example.7 page 6 Refereces page 48 Glossary page 5 Fgures pages 54-64
5 Preface My purpose wrtg ths book s to more wdely dssemate the applcatos of -admssble ad admssble permutatos to obta amlto crcuts graphs ad drected graphs. I partcular, Cojecture. hypotheszes that Algorthm G or Algorthm G o r vertces always obtas a hamlto crcut a a arbtrary graph cotag oe polyomal tme. Wder research o whether a couter-example exsts would be useful. I also would lke to kow how ofte the algorthm gve chapter, secto. gves a optmal soluto to the travelg salesma problem polyomal tme. Secodly, how ofte does t gve a approxmate soluto to the TSP that s very close to a optmal oe.
6 Itroducto Chapter. Let C be a crcle. Assume that equally-spaced pots have bee placed o the crcle at the pots π π ( =,,..., ) wth π at o'clock. Now place the umber o the crcle at the pot. Assume that we have represeted the vertces,,..., of a graph or dgraph, G, o the crcle as the cosecutve pots of a -cycle, = (... ), where the edges (arcs) of do't ecessarly le G. Call a -cycle (a b c) -admssble f (a b c) = ' where ' s also a -cycle. I theorem., we prove that (a b c) s -admssble f ad oly f the chords (a, (b)) ad (b, (c)) properly tersect C. If G has edges that have bee radomly chose, the the probablty, P, that (a b c) s -admssble s - ( - ). I partcular, as, the P. Now cosder the permutato (a c))(b d)) where (a, (c)), (c, (a)),(b, (d)), (d, (b)) are all proper, dstct chords C. (a c)(b d) s -admssble f (a c)(b d) = " where " s also a -cycle. A -admssble product of two dsjot cycles s called a POTDTC. Theorem. proves that ecessary ad suffcet codtos for (a c)(b d) to be -admssble are that a, b, c ad d are dstct pots such that the chords (a, c) ad (b, d) properly tersect the teror of C. Furthermore, f the edges of the graph G are radomly chose, the the probablty, P,, that (a c)(b d) wll be -admssble s - ( - ). Thus, as, P,. A pseudo-hamlto crcut s a -cycle ot all of whose edges belog to G. A vertex, v, G such that (v, (v)) does't belog to G s a pseudo-arc vertex. A pseudo-arc s a arc of whose tal vertex s a pseudo-arc vertex. Let G be a arbtrary graph. I the above dscusso, we assumed that s a pseudo-hamlto crcut; a, a pseudo-arc vertex of (a b c); a ad b are both pseudo-arc vertces of (a c)(b d). The -degree, d (v), of a vertex v of a drected graph D s the
umber of arcs termatg v. The out-degree, d + (v), of v s the umber of vertces emaatg from 7 v. The total degree of v equals d (v) + d + (v). If G s a graph, the the total degree of v s d(v). The mmum degree, δ(g), of a graph or dgraph s m { total degree of v}.let G cota the path P = [a, b, c,..., r] such that all of the "teror" vertces b, c,..., q have total degree. Suppose that we replace P by the vertex, v abc...qr, such that we've deleted the edge [a, r] (f t exsts). Assume that we've doe ths for every path smlar to P. The the ew graph we've obtaed s the cotracted graph of G'. I partcular, f G cotas a hamlto crcut, the δ(g'). If D s a dgraph, the both the mmum -degree ad the mmum out-degree s always greater tha the cotracted dgraph, D'. Let G be a graph whch all edges have bee radomly chose. We prove theorem.6 that f a vertex, v, of G' s of degree ad (v, (v)) s ot a edge of, the the mmal probablty, p (), that there exsts at least two -admssble -cycles cotag v s 6 5 4 86 46 + 489 8546 + 94 4 764 6 5 4 6 54 + 96 898 + 564 84 + 584 As, p () 4. Correspodg theorems are proved for radom dgraphs (drected 8 graphs). I Secto., we gve prelmares such as theorems ad deftos useful costructg the algorthms, Algorthm G (for graphs) ad Algorthm D (for dgraphs) Secto.4. Before dscussg., we gve the followg deftos: () Let be a pseudo-hamlto crcut of a graph, G, whle s s a -admssble -cycle or POTDTC. Suppose that s =. Gve that we have a table gvg the ordal values of begg wth ORD() =, we ca represet by a abbrevato cosstg of o more tha three umbers. I geeral, we use abbrevatos utl we reach oe cotag out ad costruct ew ordal umbers, ORD(), as well as ther verses, gve a vertex, we ca obta ts ordal value to use testg for had, we ca obta the uque vertex that has ORD() as ts ordal value..5 vertces. We the explctly wrte - ORD () (). Thus, -admssblty. O the other
() Suppose that S = [a, b, c,..., v, w] s a subpath of where (a, v) s a edge of G. The a rotato 8 wth respect to a ad v s a path, R = [a, v, u,..., c, b, w] that replaces S. I [9], Erd ö s ad Rey gve codtos uder whch a radom graph s -coected. I [], Pal á st gves codtos for a drected graph to be strogly-coected. I [8], K ö mlos ad Szemer é d prove that f a radom graph, G, s costructed by radomly choosg edges from the complete graph, utl every vertex has a mmum degree of at least two, the G cotas a hamlto crcut wth probablty approachg as. A smlar theorem was prove by Bollob á s [4]. I [], Freze proved a aalogous theorem for radom drected graphs. The radom graph costructed by K, Bollob á s [4], we call a Boll graph; that costructed by Freze, a Freze-Boll dgraph. Let D -, -out be a radom drected graph costructed by radomly choosg two arcs out of each f vertces ad two arcs to each vertex. Defe the radom regular -outgraph R the followg way: () Radomly select arcs out of each of vertces. () Chage each of the arcs chose to a edge. The followg cojectures are due to Freze: () D -,-out almost always cotas a hamlto crcut as. () R almost always cotas a hamlto crcut as. I [] Feer ad Freze troduced D k-,k-out ad proved that t s strogly-coected s stroglycoected for k. The hypothess of () was proved vald for k 5 by Freze ad Luczak [4]. Algorthms G ad D deped upo the ablty to costruct -admssble -cycles ad pars of dsjot -cycles (POTDTC) that pass through all of the vertces of G or D provded that G s at least -coected or D s strogly-coected. We have prevously obtaed the followg probabltes: () The probablty that a radomly chose par of arcs of the form {(a, (b)), (b, (c))} tersect the crcle C approaches as.
() The probablty that a radomly chose par of arcs {(a, (b)), (c, (d))} tersect the crcle C 9 approaches as. () If G has the property that δ (G), ad m s a arbtrary pseudo-arc vertex, the as, the mmal probablty that there wll be at least two pars of tersectos of arcs from amog those pars of form {(m, k), (k-, r)}, {(m, l), (l-, s)}, {(m, k), (, )}, {(m, k), (j, )}, {(m, l), (, )}, (m, l), ( j, )} s 4 8. ere = (,,..., m) where m s a pseudo-arc vertex of. I Secto., we formally preset Algorthm G for radom graphs G ad Algorthm D for radom dgraphs D. We frst cotract G to elmate vertces of degree two or vertces whch have -degree or out-degree less tha two dgraph D. Assume that the cotracted graphs obtaed are G' ad D', respectvely. We use a balaced, bary search tree to represet G' wth each row a brach. For D', we requre two balaced, bary search trees. The frst, ROWS, represets arcs emaatg from vertces. The other, COLUMNS,represets arcs termatg vertces. We ext obta a pseudohamlto crcut,, graph G' or D'. Let a be a arbtrary pseudo-arc vertex. We pck up to [log ] + arcs of form (a, (b)). Gve each arc chose, we costruct up to [log ] + arcs (b, (c)). We check each par {(a, (b)), (b, (c))} to see f the two arcs tersect ad thus defe a -admssble -cycle. We the select up to {[log ] + } arcs of the form (c, (a)) ad check each par of the form {(a, (b)), (c, (a))} to see f the two arcs tersect. If we are uable to obta ay par of tersectg arcs, we the test for tersecto of up to [(log ) ] pars of radomly chose edges cdet to pseudo-arc vertces a ad d (assumg that we stll have more tha oe pseudo-arc vertex). We the gve each -admssble set of vertces S = {(a, (b)), (b, (c)), (c, (a))} (or {(a, (b)), (b, (a), (d, (e)), (e, (d))}) a value called ts score. ere SCORE ( S ) = umber of arcs S - umber of arc vertces.
We choose a set, S, from amog those sets havg the hghest score, whch has a vertex of maxmum degree. Call ths set S M. If more tha oe set satsfes these codtos, we radomly pck oe of them. Let s be the permutato defed by S M. The = s. If G s a graph, ad we are uable to obta a -admssble permutato, we radomly choose a arc (a b) out of a such that the degree of b G s the greatest possble; we the use (a b) to costruct a rotato. If the termal pot of the rotato s a pseudo-arc vertex, we cotue the algorthm from there. Otherwse, we radomly choose a pseudo-arc vertex of greatest degree from PSEUDO, the set of pseudo-arc vertces that we update followg each terato. If the graph s a dgraph, we choose arcs emaatg from a or termatg (a). Sce we caot costruct a rotato out of a, we backtrack Algorthm D. If we have progressed to, + = - - s = -. We use abbrevatos to reduce the rug tme of the algorthm. I Secto.4, we dscuss the probablty of success obtag a hamlto crcut a extremal graph or dgraph as well as cojectures () ad () of Freze. I all cases, the probablty of success approaches as. I Secto.6, we prove that f a graph or dgraph cotas a hamlto crcut, C, the there always exsts a sequece of POTDTC whch lead a fte umber of steps to -admssble -cycles or C. We also prove that f a fte radom graph or radom dgraph cotas a hamlto crcut (hamlto cycle), the Algorthm G, Algorthm G o r vertces, or Algorthm D, respectvely, almost always obtas a hamlto crcut (hamlto cycle) at most.5 4 O( (log ) ) rug tme. The strkg thg about the probabltes obtaed here s that the probablty of falure decreases expoetally as the rug tme creases polyomally. Thus, f M s the umber of graphs of degree cotag hamlto crcuts, the expected umber of falures M s less tha whe the rug tme s greater tha tha a calculable umber N(M). Ths leads to the followg cojecture: Cojecture.. Let G be a arbtrary graph cotag a hamlto crcut. The Algorthm G or Algorthm G o r vertces always obtas a hamlto crcut of G polyomal tme.
I Secto.8, we gve a heurstc algorthm for obtag a approxmate soluto to the travelg salesma problem. Lastly, we gve examples of the algorthms gve chapter, Chapter. I [], Freze gves a procedure for obtag a hamlto crcut a extremal dgraph, D. s algorthm has a probablty approachg as. I the frst step, he costructs a vertex set of about log dsjot cycles that spa D. I the secod step, he patches the cycles together usg -edge exchages. I step, he patches the cycles, oe by oe, to the largest cycle by a complex process of double rotatos. The respectve rug tmes of the phases are.5 O( ) ), O((log ))), 4 +o() O( ). Before gog o, a deragemet s a permutato of the pots V = {,,...,} whch oe of the pots are fxed. Let d be a deragemet ad s a permutato such that ds = d' where d' s also a deragemet. The s s a admssble permutato. Let be a pseudo-hamlto crcut D, a radom dgraph cotag vertces such that the arcs are chose radomly utl each vertex has both -degree ad out-degree greater tha. I Secto, we gve a algorthm whch, as, a set of deragemets sequetally appled to yeld a set of approxmately log dsjot cycles spag D. The rug tme of ths algorthm s O( (log ) ). Thus, the rug tme of Freze's algorthm s reduced to 4 +o() O( ). Chapter. The frst part of ths chapter obtas a soluto to the Assgmet Problem, say σ APOPT, usg admssble permutatos ad a varat of the Floyd-Warshall algorthm gve by Papadmtrou ad Stegltz [6]. The secod part uses aother varat of F-W to obta o-egatvely-valued cycles, s ( =,,..., r ) such that σ APOPT ss... s r = σ FWTSPOPT, a approxmato to a optmal soluto to the Travelg Salesma Problem. We also gve a suffcet codto for the approxmato to be a optmal soluto. Let M be a X cost matrx. We sort each row of M creasg order of etres. We the place the colum whch the etry appears M place of ts etry. The matrx
obtaed ths maer s called MIN(M). We ext radomly costruct a deragemet, say D. We the apply D to the colums of M to obta - D M. Ths matrx has the property that ts dagoal elemets correspod to the row costs of D,. e., f (a, D(a)) = 7 M, the the etry (a, a) has the value 7 - D M. Furthermore, f (a b c) s a cycle correspodg etres M. We thus ca obta admssble permutatos value of the arc (a, b) of - D M, the (a, D(b)), (b, D(c)), (c, D(a)) are - D M. We defe the - D M wth that of the etry (a, D(a)) of M. I geeral, we wll be usg the frst colum etres of MIN(M) uless they already belog to the curret deragemet wth whch we are workg. I that case, we pck the secod colum etry. W. l. o. g., suppose that we are tryg to fd a admssble permutato, s, such that D = Dshas a smaller value tha D,.e., the sum of the values of the arcs of D s less tha the correspodg sum for D. I order to ascerta whether the cycle s wll decrease the value of D, we defe the DIFF fucto. I geeral, DIFF(a) = d(a, MIN(M)(a, )) - d(a, D(a)). Thus, DIFF( a ). We assg DIFF( a ) to each vertex a of D. I Phase, our goal each terato s to fd a permutato, s, whose value s the smallest egatvely-valued amog all tested. We the defe D = Ds Let a be the vertex of D wth the smallest egatve DIFF value. We choose the frst, secod,..., ([log ] + )-th smallest etres row a of MIN(M) to obta the smallest egatvely-valued permutato s. If we ca obta o egatvely-valued permutato, we go back to the DIFF values of the vertces of D ad test the ext [log ] smallest egatvely-valued DIFF values. If we stll caot obta a egatvely-valued permutato, we go o to Phase. Assume that we have obtaed the deragemet Before cotug, we state theorem. ad ts corollary. D Phase. Theorem. Let C be cycle cotag arcs. Assume that each arc ( a, b) has bee assged a real value, d( a, b). The f = V = d(,c()) < = there exsts at least oe value = ', wth such that
j=m j= d(' + j, C(' + j)) < where m =,,..., - ad ' + j s represeted by ts value modulo. Corollary.. Suppose that C s a cycle cotag arcs such that W = = = d(, C()) < N The there exsts a value = ' such that each partal sum S m has the property that always holds. We ow cosder S m = j=m j= d(' + j, C(' + j)) < N - D M. We frst subtract d( a, a ) from each etry d( a, j) (j=,,...,) of row a where a rus through,,...,. Ths yelds D M. Startg wth colum, we search for a egatvely-valued etry, say (, j). Usg MIN(M), we check to see f () d(, j) + d(j, k) <, () d(, j) + d(, k) < d(, k). If both codtos are satsfed, we substtute d(, j) + d(j, k) for d(, k). Every tme we obta a ew addto (, k) to a path of egatve value, P, such that P + d(, k) s stll egatve, we place the ew arc a Xmatrx called PAT. After each substtuto, we check to see f d(, k) + d(k, ) s egatve. If t s, we have a egatve cycle. If d(, k) + d(k, ) s egatve, we ca always recostruct the cycle drectly from PAT. Oce we have passed through a ew d(, k), we wrte ts egatve value talcs. Also, f a egatve path ca o loger be cotued, we wrte ts last (ad smallest) egatve value talcs. We go through colums,,..., cosecutvely. If we obta a ew egatve value at some etry (, k) where k s less tha the curret colum j' that we are workg o, we uderle ts value. Ths dcates that we caot further exted the path defed by (, k) usg the remag - j' colums. If we have goe through all colums from j = to j =, we have fshed oe terato of ths porto of the algorthm. If we have obtaed a X egatve cycle, say C, Let
D + = DC. We ca obta D + M by applyg the acto of - C to the colums of D M. If t 4 requres more tha oe terato to obta a egatve cycle, we work oly wth the uderled etres. We ca add oly oe arc obtaed from D M to each uderled egatve path durg each of the secod, thrd,... teratos. Ths porto of the algorthm cotues utl there exsts o egatve path that ca be exteded. Suppose that we have obtaed egatve cycles S = { D +r = D C C...C r. The C =,,...,r} where D +r s a optmal soluto to the Assgmet Problem,.e., σ = D. Ths sgals the ed of Phase of the algorthm. I Phase of the algorthm, we APOPT +r wsh to obta a approxmato to a optmal tour of M. Sce we use a varat of the usual F-W algorthm, we ame ths approxmato σ FWTSPOPT frst thg we must do s to obta a upper boud for of, whle a optmal tour s deoted by σ TSPOPT. The σ TSPOPT. We do ths by checkg to see f ay D + j (j=r,r-,r-,,) s a -cycle. If a -cycle s obtaed, the frst oe obtaed s a upper boud for σ FWTSPOPT. If o -cycle s obtaed, we go back to the trals that yelded D from D -. We the test to see f ay of the egatve cycles obtaed (other tha the smallest egatvely-valued oe whch gave us D ), say C', has the property that D-C'' s a -cycle. If ot, we go back to D - ad follow the same procedure. If ecessary, we go back to D tself : D s a -cycle. If oe of D + j (j=,,...,r) s a -cycle, we repeat Phase tmes. We the choose whchever tour obtaed has the smallest value. W.l.o.g., assume that D* s the tour of smallest value that we've bee able to obta. Furthermore, let m = D* - σ APOPT. Oce we have obtaed a optmal assgmet, D +r, ( D +r ) M cotas o egatve cycles. Thus, we ca use corollary. to obta cycles, T, T,..., T s each subpath of whch has value o greater tha m. Each tme we obta a ew cycle, we test to see f some subset of the cycles obtaed, say p, D+r p s a tour, T. If t s, we replace m by m = T - σ APOPT. We the obta cycle each of whose subset of arcs s o greater tha m value. We cotue the algorthm obtag m (=,,...) utl we ca o loger obta a cycle less tha m '
for some atural umber '. The algorthm cotues through at most - teratos. The smallest tour 5 obtaed up utl that pot s desgated as σ FWTSPOPT. The followg corollary of Theorem. gves a suffcet codto for σ TSTOPT = σ FWTSPOPT. Theorem.7 Suppose that m ' s the last upper boud obtaed Step a. Let σ TSPOPT = σ APOPT s where C = (a a a... ar b) be a arbtrary dsjot cycle of s where a s a determg vertex of C. The there always exsts a cycle of value o greater tha m ' obtaed Step (a) that s of the form C' = ( a b b... br' b ). Corollary. If we ca obta o cycle Step a less value tha m, the σ = σ. TSPOPT FWTSPOPT
Chapter 6 -admssble Permutatutos ad the CP. oward Klema owardklema@qcc.cuy.edu. Itroducto Let h ad h' be two -cycles S, the symmetrc group o pots. Cosder the permutato s = h h'. Sce h ad h' both have the same umber of pots, s s a eve permutato,.e., a permutato cotag a eve umber of cycles of eve legth. From elemetary group theory, every eve permutato ca be represeted as a product of (ot ecessarly dstct) -cycles. Let G be a radomly chose graph cotag vertces. Assume that G cotas a hamlto crcut,.e., a cycle made from arcs of G cotag each vertex of G precsely oce. Call t C. O the other had, let h be a radomly chose -cycle S, whle s a correspodg cycle whose arcs le K, the complete graph o vertces. s a pseudo-hamlto crcut of G. A -admssble permutato, s, s a permutato represetable as the product h h where h ad h are -cycles S.ad cotas at least as may arcs as G. Note that f (a b c) s a -admssble -cycle, the (a) = b, (b) = c, (c) = a. Thus, the acto of (a b c) trasforms the arcs (or pseudo-arcs) (a, (a)) to (a, (b)), (b, (b)) to we prove that there exsts a sequece of (b, (c)), (c, (c)) to (c, (a)). I theorem., -admssble permutatos, s, ( =,,...,r ) such that s =,..., =. + r C Copyrght
ere each s s ether a -cycle or a product of two dsjot -cycles. If e = (a, (a)) s a arc of 7 that les K G, e s a pseudo-arc of ad a s a pseudo-arc vertex. Theorem. s a exstece theorem. I theorem.6, we prove that the probablty that a radom graph of mmal degree cotas fewer tha two -admssble -cycles havg a pseudo-arc vertex s p = + + 6 5 4 86 46 489 8546 94 4 764 6 5 4 6 54 + 96 898 + 564 84 + 584. As, ths probablty of success approaches 4. We ca elmate all vertces of degree two 8 G by costructg the cotracted graph of G, G', where δ(g'). The cotracted graph s costructed by deletg all edges of G cdet to vertces of degree. Furthermore, f P = [ a,v,v,... v r,b ] s a path G such that v,v,...,v r are all of degree, f the edge [ ba], exsts, we delete t whe costructg G '. If D s a drected graph, a vertex, v, whose -degree=outdegree equals oe, s equvalet to a vertex of degree G. We say that the total degree of v s two.if P D = [ a,v,v,...,v r,b ] s a path D correspodg to P, we must delete all arcs emaatg from from a or termatg b whle costructg reasoable questo s: Are G ' ad D '. If arc (b,a ) exsts, t must also be deleted. A D ' radom? I the usual sese, o. Cosder the sets E ad N where E s the set of edges of degree removed from G durg the costructo of elemets of N are the remag edges of G removed from G whle costructg G ', whle the G '. The elemets of E have the property that each edge les o every hamlto crcut G. O the other had, N cotas edges of G each of whch les o o hamlto crcut G. owever, sce every edge of E s part of a r -vertex, a arc of t mplctly les o every pseudo-hamlto ad hamlto G. O the other had, o arc of a edge of N les o a hamlto crcut of G ' for G '. Thus, testg a set, -admssblty, o arcs that ca t le o a hamlto crcut of G are S ' of arcs S '. The same s ot true of a correspodg set of arcs G. It follows that the probablty that the frst set of arcs forms a ' -admssble permutato G ' should be as great or greater tha that of a correspodg set S
G. A correspodg argumet ca be made for D ad D '. Usg G', we use Algorthm G to 8 almost always obta a hamlto crcut. A alterate algorthm, G o r-vertces, does t requre a cotracted graph G'. I ths algorthm, let a be a pseudo-arc vertex of degree. The the probablty of cludg a edge [a b]of G a pseudo-hamlto crcut s. If G cotas a hamlto crcut, we ca almost always obta a fte sequece of -admssble permutatos, s, such that C' s a hamlto crcut G. We further prove that the ablty to create a sequece of form,,...,,... s a ecessary ad suffcet codto for the exstece of a hamlto crcut G. The algorthm used s called Algorthm G. A aalogous algorthm for dgraphs,.e., graphs whch each edge s gve a oretato, s called Algorthm D. I cotrast to Algorthm D where backtrackg s ecessary, f we fal to obta a -admssble permutato G, we use a rotato from a pseudo-arc vertex. The.5 4 rug tme for all three algorthms so( (log ) ). As we crease the rug tme of the algorthm polyomally, say by multplyg t by, we decrease the expected value of falure expoetally. From ths observato, we make the followg cojecture: Cojecture. Let G be a graph cotag a hamlto crcut. The Algorthm G or Algorthm Go r vertces always obtas a hamlto crcut at most 5 O( ) rug tme. Cojecture. Let D be a dgraph cotag a hamlto cycle.the Algorthm D always obtas a hamlto cycle at most 5 O( ) rug tme. These are dffcult cojectures to prove. If couter-examples exst, perhaps they mght help mprove the algorthms. It mght be ecessary to test all arcs out of each vertex. I the latter case, f a couterexample exsts, we should be able to obta t. Theorems 5 O( ) rug tme. We call the type of radom graph costructed by Bollabás a Boll graph.
A aalogous theorem for drected radom graphs, theorem Freze-Boll, was prove by Freze []. 9 eceforth, we assume that the set of vertces of each graph or drected graph s V. We defe the radomess of our choce of edges as Boll does [4]: Theorem.. Let G be a radom graph (radom drected graph) cotag a pseudo-hamlto crcut (cycle). The the probablty that a pseudo--cycle s -admssble s - ( - ) Proof. Let be a pseudo-hamlto crcut represeted by equ-dstatly-spaced pots traversg the crcle a clock-wse maer. Now costruct a radom chord (,(j)) whch represets a oreted edge or arc of G. ere < j. Thus, we caot let j =, sce (,) les o the crcle. Thus, the probablty, p, that (j) s chose as the termal pot of the arc s. Next, radomly costruct a arc, (j, (k)). It s easly show by costructg hσ that S = {(, (j)), (j, (k))} defes a -admssble pseudo--cycle, s = ( j k), f ad oly f the two arcs tersect the crcle. The probablty, p, that they tersect s (Pr(We radomly choose (, (j)).)( Pr((j, (k)) tersects (j).) We are assumg that o arc chose s a arc of the drected hamlto crcut. Thus, f m = - ad j' = j-, p j= j ( )( ) = ( ) ( ) = = ( ) ( ) j= The followg theorem of W. oeffdg s gve [6]: Theorem.. Let B(a,p) deote the bomal radom varable wth parameters a ad p wth BS(b,c;a,p) = Pr(b B(a,p) c) The () () BS(,( α)ap; a, p) exp(- α ap/) BS(( + α)ap, ; a, p) exp(- α ap/)
where < α <. Theorem. Let v be a radomly chose vertex of a Boll radom graph, G m*. The the probablty that v has precsely two edges of G m* cdet to t s at most log(c(log ) ) Proof.. We frst defe hypergeometrc probablty. Cosder a collecto of N = N + N smlar objects, N of them belogg to oe of two dchotomous classes (say red chps), N of them belogg to the secod class (blue chps). A collecto of r objects s selected from these N objects at radom ad wthout replacemet. Gve that X ε N, x r, x N, r x N, fd the probablty that exactly x of these r objects s chose. Pr(X = x) s gve by the formula Pr(X = x) = N N x r x N r where x objects are red ad r - x objects are blue. Let v be a radomly chose vertex of G m. We wsh to obta the probablty that exactly two edges G m are cdet to t. Let N = N + N where N s the sum of the degrees of all vertces K. r equals twce the mmum umber of edges G m for whch. Thus, N = = (-) N = the umber of edges K cdet to v = N = N - N = (-) ( ) = (-)( -) K ad N s the degree of v G m s -coected as r = the sum of the degrees of the vertces G m *
= at most [log(c(log ) )] x = I the defto of r, we assume that c s a postve umber. The Pr(X = ) = - -+ [log(c( log) )] - - [log(c(log) ] From W. Feller [], usg the approxmato of the hypergeometrc dstrbuto to the bomal dstrbuto whe N, let p = = N yeldg Pr(X = ) B(;N,p) = [ log( c(log ) ] [ log( c(log ) )] [.5log (c(log ) )]exp(-log(c(log ) )) [log(c(log ) ] c(log ) cocludg the proof. Corollary.. The probablty that G cotas more tha O(log ) vertces of degree approaches as. Proof. Usg oeffdg s Theorem, let p = (log( c(log ) )) c(log ), a = log(c(log ) ), c. Thus, ap = (log(c(log ) )). We ow smplfy ap. c(log )
log(c(log ) ) = log c + log + log(log ). But log c + log + log(log ) log as Thus for very large, ap log(c(log ) ) c < log. From () of hs theorem, the probablty, p", c that ( + α )ap vertces are of degree satsfes approachg as. But for small α >, -α ((log(c(log) ) p" < exp( )), 6c(log) log ( + α )ap <, cocludg the proof. c Theorem.4. Let D m * be a Freze-Boll drected graph. The, gve a radomly chose vertex, v, the probablty that a uque arc emaates from v s o greater tha log c c as. Proof. We aga use hypergeometrc probablty. W.l.o.g., let cotag all arcs betwee ay two vertces V. The let N = the umber of arcs = ( ) D K N = N - N = (-)(-) = (-), r = the umber of arcs D m* = at most (log + k) where k. x =. Note. Sce D m* s a Freze-Boll dgraph, each vertex,, v has the property that From hypergeometrc probablty, D K be the complete drected graph - + d (v), d (v).
Pr(X = ) = - ( -) [(log + k ) - ] - Aga usg the approxmato of the hypergeometrc dstrbuto to the bomal dstrbuto as N, we obta Pr( X = ) B(;N,p) = [((log )+k)] - - - [log + log c] exp(-[log + log c]) log( c) c [((log ) +k)] - Corollary.4. Suppose that m* = (log c) where c as. The the probablty that there exst more tha ( + ) c termatg a vertex approaches as. (log ) (log( )) uque arcs of D m ' emaatg from or Proof. The probablty, p, that a radomly chose vertex, v, of D m * has a uque arc emaatg from t approaches log c c. The umber of arcs D m* s at most a = (log + k) = (log + log c) = (log c). The umber of vertces s. Thus, ap s at most (logc) c. From oeffdg s theorem.4, α (log c) BS( + α)ap, ;a,p) exp(- c ) as. The same probablty s true for the case where a uque arc termates v. Suppose that the umber of vertces s greater tha ( + )(log ) ) vertces. The c
(logc) (logc) + (logc)(log) + (log) + c = > ( )(log) c c c 4 (logc) c (logc)(log) + > (log) c (logc) clog (logc) + > log c whch s mpossble sce the left had sde approaches a costat whle the rght-had sde approaches. The probablty that a uque arc termates a vertex, v, s the same as the probablty that t emaates from v. Thus, the total umber of such arcs s at most ( + ) (log ) (log ) as c. A -admssble product of two dsjot (pseudo) -cycles (for short a -admssble POTDTC ), say s = {(a,b), (c,d)}, occurs f ad oly f the vertces a, b, c, d traverse a clockwse maer oe of the followg ways: () a c b d, () a d b c. It follows that f the vertces are placed o postos pj for j =,,...,, the, w.l.o.g., [a, (b)] ad [c, (d)] are properly tersectg chords of. Before gog further, we meto the followg results, of whch the frst three are foud Dckso [8]: () j= j= j = ( + )( ) () j= j= j = (+ )( + ) 6 () j= j= j ( + ) =
(v) j= j= j 4 = ( + )( + )( + ) 5 (v) j= j= j 5 = ( + ) ( + ) Theorem.5. Let be a pseudo-hamlto crcut of a radom graph or a radom drected graph G. Assume that G cotas vertces ad that e ad e are radomly chose edges of G ether of whch s a arc of. The the probablty that e ad e properly tersect (have o edpots commo) s. ( ) Proof. W.l.o.g., let e = (,j). Cosder the probablty, p, that (, j) properly tersects (r, s) where r j- whle j+ s. j ca rage over the doma [,]. It follows that gve a specfc value for j, the umber of tegral values of r s j-: the edge, [r, (r)], s ot permssble by hypothess; furthermore, the loop [r, r] s ot a edge of all vertces ot cotaed K. The umber of possble values for s s -j, amely, [, j]. Thus, gve a fxed value of j, the umber of successes equals (-j)(j-). It follows that the total umber of successes s j= j= ( j)( j ) O the other had, for a fxed value of j, the umber of falures equals the umber of possble values of r (j-) multpled by the umber of possbltes for s. s caot le the closed terval [-j,]. Furthermore, t must be dstct from r ad (r). Therefore, the umber of possbltes for s s j-. Therefore, the umber of possbltes for falure s (j-). It follows that, usg all values of j, the umber of possbltes for falure s j= j= ( j )
Thus, the probablty of tersecto s 6 j= j= j= j= ( j)( j ) ( j)( j ) + ( j ) Now let j = j-, m = -. The the probablty of success smplfes to j' = m j ' = mj ' ( j ') m ( m+ ) m(m+ )( m+ ) 6 mm ( + )(m m) m = = = j' = m m ( m+ ) mm ( + )( m) m m( j') j ' = = ( ) Theorem.6. Let G be a radom graph wth vertces ad δ(g) or a radom drected graph, D, cotag vertces where both + δ (D) ad - δ (D). Assume that = (... ) s a pseudo-hamlto crcut cotag oe pseudo-arc vertex,. The the probablty that we ca obta at least two -admssble permutatos cotag s at least p = 6 5 4 86 46 + 489 8546 + 94 4 764 6 5 4 6 54 + 96 898 + 564 84 + 584. Proof. As prevously doe, let be placed at twelve o'clock ad each vertex at p for =,,...,.-. To cosder the worst possble case, assume that the degree of ad G are both. Alterately, the -degree ad out-degree of ad D are, each case,. W. l. o. g., assume that cossts of arcs or pseudo-arcs gog a clock-wse drecto. Let the followg arcs exst: c = (, ), d = (j, ), a = (, k), a' = (, l), b =(k-, r), b' = (l-, s). Sce s the oly pseudo-arc vertex of
, ay -admssble permutato must cota t. We thus have the followg possbltes for sets of 7 arcs correspodg to -admssble -cycles: () {(, k), (k-, r), (r-, )},. () {(, l), (l-, s), (s-, )}, () {(, k), (k-, +), (, )}, (4) {(, k), (k-, j+), (j, )}, (5) {(, l), (l-, +), (, )}, (6) {(, l), (l-, j+), (j, )}. (7) No -admssble -cycles are formed. The followg are geerally pseudo-arcs: (), (r-, ); (), (s-, ); (), (k-, +); (4), (k-, j+); (5), (l-, +); (6), (l-, j+). Before gog o, we preset a set of formulas that wll be useful computg the umber of possbltes for varous evets to occur: () j= j= j = ( + )( ) () j= j= j = (+ )( + ) 6 () j= j= j ( + ) = (v) j= j= j 4 = ( + )( + )( + ) (v) j= j= j 5 = ( + ) ( + ) The smplest way to obta p s to obta ts complemet, p',.e., the umber of possbltes for obtag at most oe -admssble -cycle. Ths s equvalet to obtag the umber of possbltes
for at most oe of the seve gve cases to occur; alterately, the umber of cases whch at most 8 oe par of arcs tersect. We frst obta the maxmum umber of possbltes. We radomly choose two edges cdet to ad two edges cdet to. Each edge chose caot be a loop - (, ) or (, ) - or be a arc of - (, ) or (,). We thus have - vertces avalable as the termal ed of the edge. Therefore, we frst radomly choose ad j. There are ways of choosg such a par. The smaller of the two umbers s, the other, j. Smlarly there are ways of choosg k ad l. Oce we have chose the arcs (, ), (j, ), (, k), (. l), we must radomly choose (k-, r) ad (l-, s). We have ( - ) possbltes for each choce. Thus, the total umber of possble choces s ( ) = ( ) ( ) 4 4 = 6 5 4 4 + 8 48 + 44 84 + 44 4 = 6 5 4 6 54 + 96 898 + 564 84 + 584 44 (.) W. l. o. g., we ca assume that < j ad k < l. If we assume that at most oe -admssble -cycle occurs, the ether oe or else precsely oe of the seve cases s vald. Suppose that case (6) occurs. We the have (, l) tersectg (j, ) mplyg that l < j. But k < l. Therefore, k must also be less tha j. Ths mples that both (, k) ad (, l) tersect (j, ). Therefore, we would obta at least two - admssble -cycles. Thus, we may elmate case (6). Now assume that case () occurs. The (, k) tersects (, ). Therefore, k <. But < j. mplyg that (, k) also tersects (j, ). We may thus elmate case (). Next, assume that case (5) occurs. The (, l) tersects (, ). Thus, l <. But < j. Thus, l < j. Thus, we would aga have two tersectos ad, therefore, two -admssble -cycles.
Thus, the oly cases remag are (), (), (4) ad (7) whch are llustrated fgures.,.,.4, 9.7, respectvely.. Frst cosder case () or case (). Sce there ca be at most oe -admssble -cycle, f (, k) tersects (k-, r), the k j. Otherwse, we would have at least two tersectos. Smlarly, f (, l) tersects (l-, s), the l > k j. Thus, for precsely case () or case () to occur, < j k < l. We ow cosder how may ways that ca occur. Frst, suppose that < j < k < l. The the oly costructo of arcs whch oe tersect s (, ), (j, ), (, k), (, l). Thus, we must obta the umber of ways whch we ca obta four pots from amog -,. e., 4 ad multply t by ( ). The latter s the umber of ways whch we ca costruct b = (k-, r) ad b' = (l-, s). The product obtaed s 6 5 4 8 + 494 + 96 + 48 4 (.) Next, t may occur that j=k. Thus, we compute ( ) : 5 4 + 66 64 + 96 6 = 5 4 4 5 + 64 656 + 8 84 4 (.) Addg (.) to (.), we obta 6 5 4 4 + 79 + 64 96 + 96 4 (.4) Gve the umber of possbltes that < j k < l, four cases are possble: () a tersects b, but a' does't tersect b'; () a' tersects b', but a does't tersect b; () a does't tersect b, ad a' does't tersect b';
(4) a tersects b, ad a' tersects b'. If we subtract the umber of cases whch (4) occurs from (.), gve < j k < l, we obta the umber of cases where we obta at most oe tersecto of arcs (a ad b or a' ad b') whch yeld a -admssble -cycle. We ow obta the umber of possbltes that (4) wll occur. Frst, cosder the umber of values avalable for l. Sce < k, l = k + s the smallest possble value for l. The largest value that s ca take s - sce (l,s) must properly tersect (, l). It follows that - s the largest value that l ca take. Cotug, f l = k+, the s ca take the values k+, k+,..., -. Thus, the total umber of possbltes s - k -. Sce l ca vary from k+ to -, the total umber of possbltes for a = (,l) to be tersected by (l,s) s l= l= k+ l Now cosder the umber of possbltes that a = (, k) wll be tersected by b = (k-, r). r rages from k + to -. Thus, the umber of possbltes for r s - j -. O the other had, l > k j. Therefore, k vares from j to -. It follows that - assumg that j s fxed - the umber of possbltes that both a ad b tersect ad a' ad b' tersect s k= l= k= j l= k+ ( k)( l) Cotug, rages from to j -, whle j vares from 4 to - : j k < -, Thus, the total umber of possbltes s: j= = j k= l= ( k)( l) (.5) j= 4 = k= j l= k+ We compute (.4) oe summato at a tme. Thus, cosder l= l= k+ ( l). Let U = - l -. Our sum the becomes U= U= k ( k )( k) U = U =. U= k U=
Multplyg by - k -, we obta ( k )( k) (.6) k= k= j Let U = - k -. The k = - U -. If k = j, the U = - j -. If k = -, the U =. We thus obta U= U = j U (U) U U ( j) ( j) 4 ( j)( j)(( j) ) 6 = = 8 U= j U= = ( j) ( j) ( j)( j)(( j) ) + 4 (.7) Thus, our ext computatos are: j= = j ( j) ( j) ( j)( j)(( j) ) j= 4 = 4 = j= ( j )[( j) ( j)( j)(( j) )] (.8) 4 j= 4 Let U = - j. The j = - U, j - = - U -. If j = 4, the U = - 4. If j = -, U =. Thus, (.6) yelds U= ( U )[(U (U) U(U )(U )] U= 4 4 = U= 4 ( U )[(U (U ) U(U )(U )] (.9) 4 U= Multplyg out (.9) yelds 5 4 U + (+ )U + ( )U + (8 )U + (6 )U (.) 4 U= 4 U= = + + + + 88 6 5 4 6 6 95 54 957 764 996 + 6 5 4 8 7 6 78 75 + 489 + 54 7
+ + + + 96 5 4 6 8 86 4 5444 + + 4 9 696 + 46 4966 + 94 44 4 6 + 46 4966 + 94 44 + + + 48 79 = 6 5 4 6 66 9 496 + 688 + 58 8456 44 (.) As we oted earler (.4), the umber of cases whch < j k < l whe expressed as a fracto wth 44 the deomator s 6 5 4 6 84 + 474 8 + 84 776 + 576 44 (.) It follows that the umber of cases whch < j k < l ad at most oe -admssble -cycle occurs s (.) mus (.) yeldg 6 5 4 54 774 + 58 8874 94 78 + 46 44 (.) We ow cosder case (4). ere k < j < l. a tersects d, but a' does't tersect b'. Gve fxed values for, k ad j, the umber of cases where a' = (, l) such that b' = (l -, s) does't tersect a' has s gog through the values l, l,...,,. Thus, the umber of such cases s l. But l vares from l = jto l=. Thus, our frst sum s l= l (.4) l= j
Sce k < j l, k vares from to j -. Sce a = (, k) does't tersect b = (k-, r), the cases whch ths occurs are r = k-, k-,...,,. Thus, there are k - cases whch a does't tersect b. Ths leads to k= j l= (k)(l) (.5) k= l= j vares from to j -, whle j vares from 4 to -. Therefore, our fal expresso s j j (k)(l) (.6) j= 4 = k= l= j Frst, cosder l= (k)(l) (.7) l= j Let U = l -, l = U -. If l = j, the U = j +. If l = -, U =. We thus obta U= U= j+ (+ ) j( j+ ) + j j ( j)(+ j+ ) (k )U = (k)([ ]) = (k )[ ] = (k)[ ] Next, k= j k= (+ j)( j+ ) (k)[ ] Let U = k - k = U -. If k =, U = +. If k = j -, U = j. We thus obta (+ j)( j+ ) (+ j)( j+ )( j+ )( j + ) U[ ] = (.8) 4 U= j U= Now cosder ( + j)( j+ ) { }( j + )( j+ ) (.9) 4 = j = Let U = j - +. The = j - U + + j = j - U +. If =, U = j -. If = j -, U =. Therefore, our ext sum s
(+ j)( j+ ) (+ j)( j+ ) 4 4 U= j U= j { }U(j U + ) = { }[(j+ )U U ] U= U= 4 Ths yelds (+ j)( j+ ) ( j )( j) ( j )( j)(j ) { }{[(j+ ) ] } 4 6 = (+ j)( j+ )( j )( j )(4j+ 6) 4 (.) We ow smplfy (.) ad apply our last summato: j= j= 4 + + + + + + + + + 5 4 j 5j ( )j ( )j ( 5 5 6)j (6 6). Usg formula () - (v), we obta the sum of each term separately ad the add up the sums. 4 4 ( )( ) 4 + + 76 + 4 + 76 [ ] = ( )()[ ] 6 7 5( )( ) 6 9 + + 98 45 + 5 485 [ ] = ( )()[ ] 6 4 ( + + )( )( ) 6 6 7 6 [ ] = ( )()[ ] 6 4 4 4 ( )( )( ) 4 6 + 9 + + 65 [ ] = ( )()[ ] 6 6 7 ( 5 5 6 )( )( ) 6 5 5 + + [ ] = ()()[ ] 4 6 + 6 6 8 4 ( )( 4) = The sum of the expressos o the rght-had sde of each equato s
= ( ){ + + 6 + 45 + 5 485 6 + 6 + + + + 6 + 75 + 75 45 } 6 + 8 54 7 6 4 5 5 6 4 5 54 555 54 95 5 85 5 = 4 5 + 65 + 7 8 54 7 ( )[ ] + 6 6 = 6 5 4 5 + 5 4 + 5 + 5 4 6 = 6 5 4 + 6 6 + 4 + 4 57 44 (.) Subtractg (.) ad (.) from (.), we obta 6 5 4 86 46 + 489 8546 + 94 4 + 764 44 (.) It follows that the probablty of obtag at most oe -admssble -cycle s at most 6 5 4 86 46 + 489 8546 + 94 4 + 764 6 5 4 6 54 + 96 898 + 564 84 + 584 (.) Corollary.6 Let D be a radom drected graph wth δ + (D), δ (D), wth a pseudohamlto cycle of D. The the probablty of obtag at least two -admssble permutatos cotag a arbtrary pseudo-arc vertex, v, s the same as the prevous case.
Proof. All we have to do s assume that v has two arcs emaatg from t ad (a) has at least two 6 arcs termatg t. Before begg a sketch of the algorthm, we dscuss the probablty of obtag a -admssble pseudo--cycle R, a regular -out graph obtaed from the drected graph, D. Suppose that a s a pseudo-arc vertex of the pseudo-hamlto crcut,, whle " s the graph cosstg of together wth all arcs of R symmetrc to arcs of. (For the sake of argumet, we assume that each edge of R s a par of symmetrc arcs.) Now radomly chose a edge cdet to a, say lyg R - ", ad a edge cdet to b, say [b, (c)], R - " [a, (b)],. The questo the arses: May we assume that these two edges are actually chose depedetly of each other? The aswer s yes: Each of these edges was obtaed from radomly chose arcs of D. It s possble that the arcs D mght be ( a, ( b )) ad ((c),b). The crucal thg s that the probablty that the two arcs tersect approaches as. The correspodg edges form a -admssble permutato f ad oly f they tersect. Thus, as, we may assume that radomly chose edges of the form [a, (b)], [b, (c)] have a probablty of of defg a admssble -cycle. Smlarly, edges of the form [a, (b)], [c, (d)] have a probablty of of defg a admssble POTDTC. We may thus radomly choose edges from R whe applyg Algorthm G to t. These deftos may be used terchageably. If (a, b) s a drected arc a drected graph, (b, a) s a arc symmetrc to (a, b). A radom edge chose Algorthm G of the ext secto s always assumed to be cdet to a fxed vertex, say v. I that sese, t behaves lke a arc emaatg from v. Let h = (a a... a ) deote a -cycle of S. Suppose we apply a -admssble -cycle, s, to h to obta h = hs where s = (a b c). The h' = ( ah( b)... ch( a)... bh( c)...).
Sce we have omtted oly subpaths belogg to h, 7 a' = [a,(b),...,c(a),...,b(c),...] s a abbrevato of '. Let be the pseudo-hamlto crcut correspodg to the -cycle h S, whle σ s the pseudo--cycle correspodg to s. Thus, the term -admssble s used terchageably wth h-admssble. I partcular, the pseudo-hamlto crcut,, correspodg to h ad represeted by ca be replaced by the abbrevato represetg '. As a example, f Thus, h ' ca be represeted by the abbrevato A = [a,(a),...,b,(b),...,c,(b),...] A' = [ a,( b ),...,c,( a ),...,b,( c ),...] h = ( 4 5 6 7 8 9 ), s = ( 4 8), h' = hs = ( 5 6 7 8 4 9 ) = ( h( 4 )... 8 h()... 4 h( 8 )...) a' = ( h( 4 )... 8 h()... 4 h( 8 )...) Correspodgly, sce the remag pots of all occur the order whch they occurred, the pseudo-hamlto crcut, ', s completely determed by the abbrevato A' = ( ( 4 )... 8 ()... 4 ( 8 )...). I partcular, s maps a( a ) to a( c ), b( b )to b( c ) ), c( c ) to c( a )where a =,b = 4,c = 8. Essetally, we are parttog to three subpaths that are joed together to form the pseudo-hamlto crcut '. I ths example, s s the permutato assocated wth the abbrevato A. Now let s" = ( 6)( 7) be appled to h to obta h" = hs" = (456789) = ( h(6 ) 8 h( 4 ) 6 h( ) 4 h( 8 )...)
ere we are parttog h to four subpaths joed together to form h". I geeral, the format for a 8 abbrevato usg a h-admssble POTDT, s" = (a c)(b d), s Before gog o, we defe a rotato. Let hs" = ( a h( c )... d h( b )... ch( a )... b h( d )...). S = [v,v +,v +,...,v +k ] be a subpath of a pseudo-hamlto crcut,, where v s a pseudo-arc vertex of. Assume that [v,v + k ] s a edge lyg G. The S' = [v,v +k,v +k-,...,v +] s a subpath of a pseudo-hamlto crcut, ', where S ' replaces S of to yeld '. Ths procedure s called a rotato wth respect to v ad v + k. Bollobás, Feer ad Freze use rotatos [5]. Before, gog o, we ote that each rotato defes a -admssble permutato. Let R defe pseudohamlto-cycle after the applcato of the rotato R to. The there exsts a -admssble permutato r such that r =. Wthout loss of geeralty, let R = ( ). If R s defed by the arc ( m ), the. r = ( m m... )( m m... 4 ) O the other had, ( m+ ) yelds r = ( m m... m + m... ). I partcular, f m =, the r s ( )( 4), or ( ), respectvely. Thus, whe the SCORE values of all -admssble permutatos are, rotatos may exst whose SCORE values are or. Theorem.7 deals wth these cases. If S = [v,v,...,v r ] s a subpath of G cotag teror vertces v,..., v r each of whch s of degree, the vv...v r s a r-vertex of the cotracted graph, G '. Whe costructg G ', we also delete the edge [ vv] (f t exsts). Our reaso for dog ths s that [ vv] r les o o hamlto crcut of G. I a rotato, the order of the vertces of a r-vertex s chaged. Thus, f a r-vertex s vv...v r, after a rotato t becomes v v r r...v. I costructg the cotracted graph of D, let
P = [ v,v,...,vr] be a path D where each v ( < < r) has total degree two. If r =, the the 9 -degree of v s oe.whe costructg D ', we must delete all arcs cdet to v, v,..., vr, as well as all arcs emaatg from v or termatg v r to form the r -vertex vv... v r. Furthermore, f vv...v rs a r-vertex G ' ad vrv r+...v s s a s +-vertex, we must combe them to vv...v rv r+...v s, a (r + s)-vertex, that the becomes a vertex G'. Before cludg ths (r+s)-vertex G', we must delete all edges of G cdet to v r. A reasoable questo s: Are G ' ad D ' radom? I the usual sese, o. Cosder the sets E ad N where E s the set of edges of degree removed from G durg the costructo of removed from G whle costructg G ', whle the elemets of N are the remag edges of G G '. The elemets of E have the property that each edge les o every hamlto crcut G. O the other had, N cotas edges of G each of whch les o o hamlto crcut G. owever, sce every edge of E s part of a r -vertex, a arc of t mplctly les o every pseudo-hamlto ad hamlto G. O the other had, o arc of a edge of N les o a hamlto crcut of G '. Thus, testg a set, ca t le o a hamlto crcut of G are S ' of arcs G ' for G. It follows that the probablty that the frst set of arcs forms a -admssblty, o arcs that S '. The same s ot true of a correspodg set of arcs ' -admssble permutato should be as great or greater tha that of a correspodg set S G. A correspodg argumet ca be made for D ad D '. Next, we dscuss the relatoshp betwee G, G-, ad the crcle (( -). Frst, let =. We obta the radom graph, G, by radomly choosg m edges ad smultaeously placg them. Thus, each par of edges has probablty G ' = of beg ( -) chose, whle each vertex has probablty. The mportat fact s that these probabltes are depedet of the probabltes of each of the other edges. We ow ote that G- s geerally ot a radom graph. Wthout loss of geeralty, let = (... ). Suppose we choose the arc ( j).
Now suppose that G s a radom graph cotag the set of vertces V= {,,..., }. Let j be a 4 arbtrary vertex other tha. The the probablty that a arbtrary edge cdet to j has ts other edpot the terval [[,,..., ] s j-. Now assume that we have placed the vertces equ- - dstat from each other alog the crcle e = ( j ). Furthermore, f a arc of G les o = (,,, }. Suppose we radomly choose the arc, we recogze ths fact before choosg a arc f = ( j-k) emaatg from j-. Cosder ( j- j). If t s a edge of G, the umber of possbltes for k j s j. Thus, the probablty that k j s at most j-. O the other had, - f j-s a pseudo-arc vertex, the ( j- j) does t le G. It follows that the probablty k j s aga at most j- -. Thus, usg Algorthms G, G o r vertces, or D, the probablty that f tersects e as a chord the teror of s greater tha the probablty that ths wll occur whe arc f s chose radomly from the radom graph G. A smlar stuato occurs whe we test for -admssble POTDTC s. Furthermore, as the algorthms proceeds, the set of arcs, pseudo-arcs ad pseudo-arc vertces chages. Assume that every edge s of G s represeted by a symmetrc par of arcs. The, f vv...v rs a r-vertex of G, the same s true of vrv r...v vv...v r the same pseudo-hamlto cycle as vrv r...v. Our oly caveat s that we ca t have or vce-versa: vv...v rv r...v s a sequece of o-dsjot -cycles. Thus, they ca t both occur a -cycle. Gve vv...v r a pseudohamlto cycle, the oly way we ca obta vrv r...v s f vv... v rles o that part of a rotato whch the oretatos of the arcs that defe t are reversed. The.ew pseudo-hamlto crcut obtaed from replacg subpaths of the form S by r-vertces s '. We ote that, by costructo, the umber of pseudo-arc vertces of ' s ever greater tha the umber. Furthermore, after deletg all edges of G m* from vertces of degree two, all of the remag arcs of G m* occur ' G m*. Let t be umber of vertces of degree G m*. We remove at most t edges from G m* to form V C, the
set of vertces of the cotracted graph, G ' m*. As we wll prove theorems. ad., gve that a 4 graph or dgraph cotas a hamlto crcut or cycle, admssble pseudo -cycles ad POTDTC' s that yelds C, there always exsts a sequece of - C. To obta a -admssble permutato (a b c) requres that [a,(b)] ad [b,(c)] properly tersect a crcle alog whch the vertces of are equally spaced. O the other had, -admssblty of a POTDT, (a c)(b d), requres that [a, c] ad [b, d] properly tersect. We search each terato depth up to (log ) permutatos. Thus, usg abbrevatos whe testg for -admssblty, we eed oly cosder at most 4(log ) pots for the frst terato, 48(log ) pots for the secod oe,..., 48r(log ) pots for the r-th oe. Thus, r teratos, we radomly go through 4(log ) pots. Suppose a recalculato of every teratos. The (as wll be show more detal secto.4), t requres occurs O( (log ) )r. t. to go through teratos. O the other had, t requres O( ) rug tme (r. t.) to costruct a rotato a abbrevato cotag pots. It follows that t requres at most O( (log )) rug tme to use a rotato each of teratos. It follows that at most.5 O( (log ) ) teratos, t requres.5 4 O( (log ) ) r. t. to complete these operatos. The latter wll also be the r. t. for both Algorthm G ad Algorthm D the ext secto. The algorthm essetally cossts of sequetally obtag a sequece,a,... A.5,.5,A.5,...,A.5,.5,... whch, applyg [ ] [ ] [ ] + [ ] [ ] Algorthm G to graphs, the umber of pseudo-arc vertces s a mootocally decreasg fucto. Usg Algorthm D for drected graphs, we are requred to backtrack after certa teratos. owever, usg a large umber of teratos, the umber of successful teratos becomes cosderably greater tha the umber of falures. Thus, we here also replace pseudo-arc vertces by arc vertces. I geeral, as, we approach a hamlto crcut the former case, ad a hamlto cycle the latter. Deftos ad examples of the followg data structures comes from Kuth [7]: Gve m etres, usg a balaced, bary search tree, we ca, respectvely, locate, sert, or delete ay elemet, or