Header Page of 123 `.I CAM D - OAN LO Toˆi xin cam d¯oan r˘`a ng c´ac keˆ´t qua’ d¯u.o c tr`ınh b`ay luaˆn ´an l`a u.ng d¯u.o c coˆng boˆ´ o’ baˆ´t k`y moˆt ho`an to`an m´o.i, chu.a t` coˆng tr`ınh khoa ho.c cu’a kh´ac H` a Noˆi, ang n˘am 2005 ng`ay th´ Tra o.c ˆ`n Minh Tu.´ Footer Page of 123 Header Page of 123 MU C LU C L` o.i cam d ¯oan Mu.c lu.c Danh mu.c c´ ac h`ınh ’ D ˆ`U -A MO ’N ´ KIE ´ C CO BA ˆ´N THU Chu.o.ng CAC 12 - oˆ` thi 1.1 D 12 - oˆ` thi b˘a´c caˆ`u d¯ı’nh v`a d¯oˆ` thi meta luaˆn ho`an 1.2 D 17 1.2.1 Nh´ om ho´ an vi 17 1.2.2 C´ac d¯.inh ngh˜ıa 19 1.3 T´ınh lieˆn thoˆng 22 1.4 B`ai to´an Hamilton 25 ˆ N THO ˆ NG CU’A D ˆ` THI -O Chu.o.ng T´INH LIE ˆC BA 2.1 Moˆt soˆ´ t´ınh chaˆ´t cu’a d¯oˆ` thi meta luaˆn ho`an 2.2 Tru.`o.ng ho p S0 = ∅ 2.3 Tru.`o.ng ho p S0 = ∅ ˆ N HOAN ` META LUA 29 29 34 41 ˆN ˆ` THI META LUA -O Chu.o.ng CHU TR`INH HAMILTON TRONG D ` BA ˆ C HOAN 66 3.1 Moˆt soˆ´ boˆ’ d¯ˆe` 66 - ieˆ`u kieˆn 3.2 D d¯u’ cho su toˆ`n ta.i chu tr`ınh Hamilton 73 ˆN ˆ´T LUA KE 82 Danh mu.c c´ ac co ˆng tr`ınh 83 T` lie ˆu tham kha’o 84 Footer Page of 123 Header Page of 123 ´ ` DANH MU C CAC HINH 1.1 Bieˆ’u dieˆ˜n d¯oˆ` thi treˆn m˘a.t ph˘a’ ng - oˆ` thi ca’m sinh G v`a d¯oˆ` thi bao tr` 1.2 D um G cu’a G 14 1.3 Hai d¯oˆ` thi d¯a˘’ ng caˆ´u G v`a G 15 1.4 Baˆc cu’a d¯ı’nh, baˆc cu’a d¯oˆ` thi 1.5 V´ı du d¯oˆ` thi d¯ˆe`u 16 1.6 C´ac d¯oˆ` thi b˘a´c caˆ`u d¯ı’nh Coxeter (G1 ) v`a Petersen (G2 ) 19 - oˆ` thi luaˆn ho`an 1.7 D 20 - oˆ` thi meta luaˆn ho`an 1.8 D - oˆ` thi G v´o.i chu tr`ınh C v`a d¯u.`o.ng P 1.9 D - oˆ` thi v´o.i c´ac th`anh phaˆ`n cu’a n´o 1.10 D - oˆ` thi Hamilton v`a nu’.a Hamilton 1.11 D 21 cu’a n´o 23 24 25 - i.nh l´ 3.1 V´ı du minh ho.a cho D y 3.7 - i.nh l´ 3.2 V´ı du minh ho.a cho D y 3.9 76 78 - inh l´ 3.3 V´ı du minh ho.a cho D y 3.10 80 Footer Page of 123 13 16 Header Page of 123 - ˆ` AU MO’ D Luaˆn lieˆn thoˆng v`a su toˆ`n ta.i chu tr`ınh a´n d¯ˆe` caˆp t´o i d¯ieˆ`u kieˆn - ´o l`a moˆt u.ng Hamilton cu’a c´ac d¯oˆ` thi meta luaˆn ho`an baˆc D nh˜ l´o.p d¯oˆ` thi b˘a´c caˆ`u d¯ı’nh c`on ´ıt d¯u.o c quan taˆm xem x´et moˆt u.u nhieˆ`u soˆ´ l´o.p d¯oˆ` thi b˘a´c caˆ`u d¯ı’nh kh´ac, gaˆ`n d¯ˆay, d¯a˜ d¯u.o c nghieˆn c´ L´y thuyeˆ´t d¯oˆ` thi d¯a˜ d¯u.o c h`ınh th`anh t` u laˆu v`a c´o u ´.ng du.ng roˆng ´ tu.o’.ng co ba’n cu’a r˜ai nhieˆ`u l˜ınh vu c khoa ho.c v`a thu c tieˆ˜n Y l´y thuyeˆ´t d¯oˆ` thi d¯a˜ d¯u.o c nhieˆ`u nh`a khoa ho.c d¯ˆe` xuaˆ´t v`ao nu’.a d¯aˆ`u theˆ´ ky’ 18 Tieˆu bieˆ’u l`a Leonhard Euler (1707 – 1783), nh`a to´an ho.c u.u b`ai to´an “Ba’y caˆy caˆ`u o’ noˆ’i tieˆ´ng ngu.`o.i Thu.y S˜ı, oˆng nghieˆn c Kăonigsberg - o` thi l`a mot D uc toan ho.c r`o.i ra.c bieˆ’u dieˆ˜n moˆ´i quan heˆ gi˜ u.a caˆ´u tr´ u.c, ta c´o theˆ’ h`ınh dung moˆt c´ac d¯oˆ´i tu.o ng Moˆt c´ach phi h`ınh th´ d¯oˆ` thi bao goˆ`m c´ac “d¯ı’nh” v`a c´ac “ca.nh”, moˆ˜i ca.nh noˆ´i moˆt c˘a.p d¯ı’nh n`ao d¯o´ uc d¯oˆ` thi Nhieˆ`u b`ai to´an thu c teˆ´ c´o theˆ’ d¯u.o c moˆ h`ınh ho´a b˘a` ng caˆ´u tr´ Ch˘a’ ng ha.n, thieˆ´t laˆp u.a c´ac th`anh phoˆ´ cu’a moˆt tuyeˆ´n bay gi˜ quoˆ´c ung gia th`ı d¯oˆ` thi gi´ up ch´ ung ta so d¯oˆ` ho´a heˆ thoˆ´ng n`ay b˘a` ng c´ach d` moˆ˜i d¯ı’nh bieˆ’u thi moˆt th`anh phoˆ´ c`on moˆ˜i ca.nh bieˆ’u dieˆ˜n moˆt tuyeˆ´n ´.ng; moˆt bay th˘a’ ng gi˜ u.a hai th`anh phoˆ´ tu.o.ng u v´ı du kh´ac: thieˆ´t keˆ´ ma.ch in cho moˆt “bo” ma.ch d¯ieˆn tu’ , nhieˆ`u keˆ´t qua’ veˆ` d¯oˆ` thi ph˘a’ ng s˜e gi´ up ta t`ım d¯u.o c moˆt so d¯oˆ` thieˆ´t keˆ´ hieˆu qua’ Nhu vaˆy, u.u caˆ´u tr´ uc cu’a nh˜ u.ng l´o.p d¯oˆ` thi kh´ac vieˆc nghieˆn c´ - ˘a.c bieˆt c` ung v´o.i c´ac u ´.ng du.ng cu’a n´o l`a h˜ u.u ´ıch D l`a th`o i d¯a.i ng`ay nay, coˆng ngheˆ thoˆng tin v´o.i voˆ soˆ´ qu´a tr`ınh xu’ l´y v`a truyeˆ`n u.u tin d¯ang thaˆm nhaˆp soˆ´ng th`ı vieˆc nghieˆn c´ v`ao mo.i l˜ınh vu c cu’a cuoˆc u.ng nghieˆn c´ u.u l´y thuyeˆ´t n`ay la.i c`ang c´o y ´ ngh˜ıa Ngu.o c la.i, nh˜ Footer Page of 123 Header Page of 123 u.ng keˆ´t qua’ m´o.i saˆu s˘a´c ho.n nh`o su tieˆ´n boˆ cu’a d¯oˆ` thi s˜e d¯a.t d¯u.o c nh˜ khoa ho.c m´ay t´ınh V´o.i moˆt d¯oˆ` thi cho tru ´o c, t´ınh lieˆn thoˆng cu’a n´o thu `o ng d¯u o c quan taˆm d¯aˆ`u tieˆn Ch˘a’ ng ha.n, moˆ h`ınh cu’a moˆt heˆ thoˆ´ng giao thoˆng nhaˆ´t - ˜a c´o nh˜ thieˆ´t pha’i l`a moˆt u.ng thuaˆt u.u d¯oˆ` thi lieˆn thoˆng D to´an kh´a h˜ hieˆu d¯ˆe’ kieˆ’m tra t´ınh lieˆn thoˆng cu’a moˆt d¯oˆ` thi., nhu ng caˆu tra’ l`o i o’ d¯o´ m´o.i chı’ l`a “C´o” ho˘a.c “Khoˆng” lieˆn thoˆng V´o.i nhieˆ`u l´o.p d¯oˆ` thi cu theˆ’, c´ac nh`a nghieˆn c´ u.u thu.`o.ng mong muoˆ´n c´o moˆt kh˘a’ ng d¯.inh ma.nh ho n Do vaˆy, l´o p d¯oˆ` thi n`ao d¯o´ vaˆ´n d¯ˆe` d¯a˘ c tru ng t´ınh lieˆn thoˆng cu’a moˆt - ieˆ`u n`ay khoˆng pha’i l´ uc n`ao c˜ ung c˜ ung thu.`o.ng d¯u.o c d¯u.a xem x´et D nhaˆn d¯u o c deˆ˜ d`ang Chı’ c´o moˆt soˆ´ keˆ´t qua’ cu’a Menger (1927) v`a Tutte (1961) veˆ` d¯oˆ lieˆn thoˆng (connectivity) cu’a moˆt d¯oˆ` thi (xem [13]) u.ng l´o.p he.p ho.n V`ı theˆ´, ngu.`o.i ta thu.`o.ng xem x´et vaˆ´n d¯ˆe` n`ay treˆn nh˜ Moˆt u.a m`a cho t´o.i vaˆ˜n d¯ang d¯u.o c coi l`a vaˆ´n d¯ˆe` trung vaˆ´n d¯ˆe` n˜ taˆm cu’a l´y thuyeˆ´t d¯oˆ` thi l`a b`ai to´an Hamilton: V´o.i moˆt d¯oˆ` thi cho tru.´o.c, h˜ay x´ac d¯.inh xem c´o hay khoˆng moˆt h`anh tr`ınh d¯i qua taˆ´t ca’ c´ac d¯ı’nh cu’a d¯oˆ` thi., moˆ˜i d¯ı’nh d¯u ´ng moˆt laˆ`n, roˆ`i la.i quay tro’ veˆ` d¯ı’nh xuaˆ´t ph´at? H`anh tr`ınh tho’a m˜an b`ai to´an Hamilton d¯u.o c go.i l`a chu tr`ınh Hamilton Neˆ´u khoˆng yeˆu caˆ`u pha’i tro’ veˆ` d¯u ´ng d¯ı’nh xuaˆ´t ph´at th`ı h`anh tr`ınh n`ay s˜e d¯u.o c go.i l`a d¯u.`o.ng Hamilton B`ai to´an Hamilton l`a moˆt b`ai to´an l´o n, nhu ng m´o i chı’ d¯u o c gia’i quyeˆ´t cho nh˜ u.ng tru.`o.ng ho p d¯a˘ c bieˆt Do d¯o´, xem x´et b`ai to´an u.ng ha.n cheˆ´ leˆn c´ac d¯oˆ` thi d¯ˆe’ nghieˆn n`ay, ngu.`o.i ta thu.`o.ng d¯a˘ t nh˜ ung theo moˆt u vaˆy, u.ng c´ u.u ch´ c´ach tieˆ´p caˆn n`ao d¯o´ M˘a.c d` d¯a phaˆ`n nh˜ coˆng tr`ınh nghieˆn c´ u.u c˜ ung chı’ d¯u.a d¯u.o c d¯ieˆ`u kieˆn d¯u’ d¯ˆe’ moˆt d¯oˆ` thi c´o chu tr`ınh Hamilton Ch˘a’ ng ha.n, d¯.inh l´y cu’a Dirac kh˘a’ ng d¯.inh veˆ` su toˆ`n ta.i cu’a chu tr`ınh Hamilton c´ac d¯oˆ` thi c´o soˆ´ ca.nh “d¯u’ l´o.n” v`a “phaˆn boˆ´ d¯ˆe`u treˆn c´ac d¯ı’nh”, hay keˆ´t qua’ cu’a Tutte chı’ r˘a` ng c´ac d¯oˆ` thi ph˘a’ ng (d¯oˆ` thi c´o theˆ’ bieˆ’u dieˆ˜n d¯u.o c treˆn m˘a.t ph˘a’ ng cho Footer Page of 123 Header Page of 123 c´ac ca.nh cu’a n´o khoˆng c˘a´t nhau) v`a c´o su “lieˆn thoˆng ma.nh” th`ı s˜e c´o chu tr`ınh Hamilton (xem chi tieˆ´t [13], [14], [19]) - aˆy Gaˆ`n d¯ˆay, ngu.`o.i ta quan taˆm nhieˆ`u d¯ˆe´n d¯oˆ` thi b˘a´c caˆ`u d¯ı’nh D l`a c´ac d¯oˆ` thi c´o nh´om tu d¯a˘’ ng caˆ´u t´ac d¯oˆng b˘a´c caˆ`u leˆn taˆp d¯ı’nh cu’a u.a d¯ı’nh baˆ´t k`y luoˆn toˆ`n ta.i c´ac tu d¯a˘’ ng caˆ´u chuyeˆ’n ch´ ung, t´ u.c l`a gi˜ ch´ ung veˆ` Nhu vaˆy, d¯oˆ` thi b˘a´c caˆ`u d¯ı’nh l`a l´o p d¯oˆ` thi mang t´ınh u.ng t´ınh chaˆ´t l´y th´ u V´ı du., gia’ thuyeˆ´t d¯oˆ´i x´ u.ng cao neˆn c´o theˆ’ c´o nh˜ Lov´asz (1968, xem [18], [21]) cho r˘a` ng: “Mo.i d¯oˆ` thi b˘a´c caˆ`u d¯ı’nh lieˆn thoˆng d¯ˆe`u c´o d¯u.`o.ng Hamilton”, hay gia’ thuyeˆ´t Thomassen (xem [10], [18]) d¯a˜ neˆu: “Chı’ c´o moˆt u.u ha.n c´ac d¯oˆ` thi b˘a´c caˆ`u d¯ı’nh lieˆn soˆ´ h˜ thoˆng l`a khoˆng c´o chu tr`ınh Hamilton” Nh˜ u.ng n˘am tro’ la.i d¯ˆay, uy ´ t´o.i u ´.ng du.ng cu’a d¯oˆ` thi b˘a´c nghieˆn c´ u.u l´y thuyeˆ´t, ngu.`o.i ta c`on ch´ caˆ`u d¯ı’nh cho moˆ h`ınh ma.ng lieˆn keˆ´t hay c´ac heˆ thoˆ´ng xu’ l´y song song Ngo`ai ra, c´o nh´om tu d¯a˘’ ng caˆ´u t´ac d¯oˆng b˘a´c caˆ`u treˆn taˆp d¯ı’nh, u.u b˘`a ng l´y thuyeˆ´t neˆn d¯oˆ` thi b˘a´c caˆ`u d¯ı’nh khoˆng nh˜ u.ng d¯u.o c nghieˆn c´ toˆ’ ho p m`a c`on c´o theˆ’ su’ du.ng ca’ d¯a.i soˆ´ (cu theˆ’ l`a l´y thuyeˆ´t nh´om) d¯ˆe’ xem x´et ch´ ung theo moˆt g´oc d¯oˆ kh´ac - oˆ´i v´o.i d¯oˆ` thi b˘a´c caˆ`u d¯ı’nh, caˆ´u tr´ D uc cu’a nh´om c´ac tu d¯a˘’ ng caˆ´u treˆn d¯oˆ` thi d¯o´ng moˆt u.u vai tr`o quan tro.ng Tuy nhieˆn vieˆc nghieˆn c´ ´ c˜ ung khoˆng pha’i d¯oˆ` thi b˘a´c caˆ`u d¯ı’nh v´o.i nh´om c´ac tu d¯a˘’ ng caˆ´u tu`y y deˆ˜ d`ang V`ı theˆ´, ngu.`o.i ta thu.`o.ng nghieˆn c´ u.u d¯oˆ` thi b˘a´c caˆ`u d¯ı’nh v´o.i u d¯o.n gia’n d¯ˆe´n ph´ u.c ta.p nh´om tu d¯a˘’ ng caˆ´u t` - oˆ` thi luaˆn ho`an l`a d¯oˆ` thi b˘a´c caˆ`u d¯ı’nh c´o caˆ´u tr´ D uc d¯o.n gia’n nhaˆ´t: ung ch´ u.a moˆt nh´om tu d¯a˘’ ng caˆ´u cu’a ch´ nh´om xyclic t´ac d¯oˆng b˘a´c u.u nhieˆ`u nhaˆ´t caˆ`u leˆn taˆp d¯ı’nh V`ı vaˆy c´ac d¯oˆ` thi n`ay d¯a˜ d¯u o c nghieˆn c´ soˆ´ c´ac d¯oˆ` thi b˘a´c caˆ`u d¯ı’nh Treˆn c´ac d¯oˆ` thi luaˆn ho`an, b`ai to´an Hamilton v`a b`ai to´an phaˆn l´o.p d¯a˜ d¯u.o c gia’i quyeˆ´t tro.n ve.n Trong [17], ngu.`o.i ta d¯a˜ chı’ r˘`a ng d¯oˆ` thi luaˆn ho`an n˘`a m l´o.p d¯oˆ` thi Cayley Footer Page of 123 Header Page of 123 uc (xem d¯.inh ngh˜ıa o’ trang 22), moˆt l´o p d¯oˆ` thi b˘a´c caˆ`u d¯ı’nh c´o caˆ´u tr´ ung tu.o.ng d¯oˆ´i roˆng kh´a ch˘a.t ch˜e nhu.ng c˜ uc ph´ u.c ta.p L´o.p d¯oˆ` thi m`a nh´om c´ac tu d¯a˘’ ng caˆ´u cu’a n´o c´o caˆ´u tr´ u.a ho.n d¯o´ l`a l´o.p d¯oˆ` thi meta luaˆn ho`an Nh´om tu d¯a˘’ ng caˆ´u cu’a n´o ch´ b˘a´c caˆ`u leˆn moˆt nh´om g, h , sinh bo’ i hai phaˆ`n tu’ g, h, t´ac d¯oˆng ’ taˆp d¯ı’nh v`a g, h l`a t´ıch nu’ a tru c tieˆ´p cu’a g v´o i h O d¯ˆay, t´ıch nu’.a tru c tieˆ´p cu’a nh´om K v´o.i nh´om L l`a nh´om M ch´ u.a c´ac nh´om K v`a L cho K d¯a˘’ ng caˆ´u v´o.i K, L d¯a˘’ ng caˆ´u v´o.i L, K v`a L chı’ chung phaˆ`n tu’ d¯o.n vi., K l`a nh´om chuaˆ’n t˘a´c cua’ M v`a M d¯u.o c sinh bo’.i K v`a L - oˆ` thi meta luaˆn ho`an d¯u.o c d¯ˆe` xuaˆ´t v`a nghieˆn c´ D u.u d¯aˆ`u tieˆn bo’.i B Alspach v`a T.D Parsons t` u n˘am 1982 (xem [5]) Trong b`ai b´ao n`ay, c´ac t´ac gia’ d¯a˜ d¯u.a moˆt d¯.inh ngh˜ıa toˆ’ ho p cho d¯oˆ` thi meta luaˆn ho`an, ch´ u.ng minh moˆt uc cu’a c´ac d¯oˆ` thi n`ay v`a x´ac soˆ´ keˆ´t qua’ veˆ` caˆ´u tr´ u.a ba l´o.p d¯oˆ` thi luaˆn ho`an, meta luaˆn ho`an d¯.inh d¯u.o c moˆ´i lieˆn heˆ gi˜ v`a Cayley O’ d¯ˆay, moˆt d¯oˆ` thi meta luaˆn ho`an d¯u o c cho bo’ i c´ac tham soˆ´ caˆ´u tr´ uc bao goˆ`m hai soˆ´ nguyeˆn du.o.ng m, n x´ac d¯.inh soˆ´ d¯ı’nh v`a su phaˆn boˆ´ c´ac d¯ı’nh cu’a d¯oˆ` thi., soˆ´ α nguyeˆn toˆ´ v´o.i n v`a moˆt soˆ´ taˆp cu’a taˆp c´ac soˆ´ nguyeˆn modulo n, d¯u o c go.i l`a c´ac bieˆ’u tu o ng cu’a d¯oˆ` thi - ˘a.c bieˆt, meta luaˆn ho`an, x´ac d¯.inh c´ac ca.nh cu’a d¯oˆ` thi D keˆ´t luaˆn u.u cho cu’a b`ai b´ao, Alspach v`a Parsons d¯a˜ d¯ˆe` xuaˆ´t ba hu.´o.ng nghieˆn c´ u.u kh´a phoˆ’ bieˆ´n l`a vaˆ´n c´ac d¯oˆ` thi n`ay, d¯o´ c´o hai hu.´o.ng nghieˆn c´ d¯ˆe` d¯a˘’ ng caˆ´u v`a b`ai to´an Hamilton treˆn l´o.p d¯oˆ` thi meta luaˆn ho`an Theo c´ac hu.´o.ng nghieˆn c´ u.u treˆn, vaˆ´n d¯ˆe` toˆ`n ta.i chu tr`ınh Hamilton - ˜a c´o moˆt d¯u.o c quan taˆm nhieˆ`u ho.n D u.ng l´o.p d¯oˆ` thi soˆ´ keˆ´t qua’ cho nh˜ meta luaˆn ho`an d¯u.o c ha.n cheˆ´ bo’.i c´ac d¯ieˆ`u kieˆn r`ang buoˆc kh´ac Alspach v`a nh´om nghieˆn c´ u.u d¯a˜ keˆ´t luaˆn r˘`a ng mo.i d¯oˆ` thi meta luaˆn ho`an v´o.i tham soˆ´ n nguyeˆn toˆ´ v`a kh´ac d¯oˆ` thi Petersen (xem trang 19) d¯ˆe`u c´o chu tr`ınh Hamilton [4] Moˆt soˆ´ b`ai b´ao kh´ac la.i d¯ˆe` caˆp t´o i l´o p Footer Page of 123 Header Page of 123 d¯oˆ` thi Cayley Ch˘a’ ng ha.n [8], [16], [22], c´ac t´ac gia’ d¯a˜ chı’ u.ng d¯oˆ` thi Cayley treˆn c´ac su toˆ`n ta.i cu’a chu tr`ınh Hamilton nh˜ nh´om c´o caˆ´u tr´ uc d¯a˘ c bieˆt Trong d¯o´, t´ınh lieˆn thoˆng cu’a c´ac d¯oˆ` thi la.i gi˜ u moˆt vai tr`o quan ´ ngh˜ıa treˆn c´ac tro.ng d¯oˆ´i v´o.i b`ai to´an Hamilton B`ai to´an n`ay chı’ c´o y - ˘a.c bieˆt d¯oˆ` thi lieˆn thoˆng D treˆn c´ac d¯oˆ` thi cho bo’ i c´ac tham soˆ´ caˆ´u tr´ uc nhu d¯oˆ` thi meta luaˆn ho`an, ngu.`o.i ta muoˆ´n c´o d¯u.o c d¯ieˆ`u kieˆn caˆ`n v`a d¯u’ cho t´ınh lieˆn thoˆng cu’a c´ac d¯oˆ` thi n`ay Khi d¯a˜ d¯a˘ c tru.ng d¯u.o c u.ng r`ang buoˆc u.a t´ınh lieˆn thoˆng cu’a d¯oˆ` thi meta luaˆn ho`an b˘a` ng nh˜ gi˜ c´ac tham soˆ´ caˆ´u tr´ uc, vieˆc xem x´et su toˆ`n ta.i chu tr`ınh Hamilton ch´ ung s˜e thuaˆn lo i ho n Tru.´o.c thu c teˆ´ n`ay, luaˆn u.u veˆ` l´o.p d¯oˆ` thi meta luaˆn a´n nghieˆn c´ ’ ung toˆi khoˆng su’ du.ng ho`an v`a d¯oˆ` thi meta luaˆn ho`an baˆc O d¯ˆay, ch´ u.u theo tham soˆ´ nh˜ u.ng c´ach tieˆ´p caˆn tru ´o c d¯o´ m`a d¯.inh hu ´o ng nghieˆn c´ “baˆc” (xem d¯.inh ngh˜ıa o’ trang 15) cu’a d¯oˆ` thi b˘a´c caˆ`u d¯ı’nh Trong c´ac moˆ h`ınh ma.ng lieˆn keˆ´t, d¯oˆ` thi b˘a´c caˆ`u d¯ı’nh baˆc nho’ c´o moˆt ´ ngh˜ıa quan tro.ng C´ac d¯oˆ` thi b˘a´c caˆ`u d¯ı’nh baˆc y v`a baˆc c´o theˆ’ - oˆ` thi b˘a´c caˆ`u d¯ı’nh baˆc d¯u.o c moˆ ta’ d¯aˆ`y d¯u’ m`a khoˆng maˆ´y kh´o kh˘an D l`a ho p r`o.i cu’a c´ac d¯oˆ` thi K2 , c`on d¯oˆ` thi b˘a´c caˆ`u d¯ı’nh baˆc l`a ung d¯oˆ d`ai Trong soˆ´ c´ac d¯oˆ` thi ho p cu’a c´ac chu tr`ınh r`o.i v`a c´o c` b˘a´c caˆ`u d¯ı’nh baˆc 3, d¯oˆ` thi meta luaˆn ho`an baˆc ´ıt nhieˆ`u d¯a˜ d¯u o c xem ung nhu su toˆ`n ta.i x´et v`a d¯a.t d¯u.o c nhieˆ`u keˆ´t qua’ veˆ` t´ınh lieˆn thoˆng c˜ chu tr`ınh Hamilton (xem [25] – [29], [31], [33] – [36]) Moˆt c´ach tru c quan, ngu `o i ta deˆ˜ laˆ`m tu o’ ng r˘a` ng mo.i d¯oˆ` thi meta u.ng luaˆn ho`an baˆc u.a c´ac d¯oˆ` thi meta luaˆn ho`an baˆc nhu nh˜ c´o theˆ’ ch´ d¯oˆ` thi Nhu.ng thu c teˆ´ khoˆng d¯u.o c nhu ta mong muoˆ´n Do caˆ´u tr´ uc d¯a˘ c bieˆt soˆ´ ´ıt c´ac d¯oˆ` thi meta luaˆn cu’a nh´om tu d¯a˘’ ng caˆ´u, chı’ moˆt ho`an baˆc u.ng l`a d¯oˆ` thi cu’a d¯oˆ` thi meta luaˆn ho`an baˆc Do d¯o´ nh˜ Footer Page of 123 Header Page of 123 k˜y thuaˆt haˆ`u nhu d¯u o c su’ du.ng treˆn l´o p d¯oˆ` thi meta luaˆn ho`an baˆc khoˆng a´p du.ng d¯u.o c d¯oˆ´i v´o.i d¯oˆ` thi meta luaˆn ho`an baˆc V´o i hy vo.ng u.ng k˜y thuaˆt s˜e t`ım t`oi d¯u.o c nh˜ m´o i c´o theˆ’ ´ap du.ng cho ca’ l´o p d¯oˆ` thi meta luaˆn ho`an toˆ’ng qu´at, ch´ ung toˆi d¯a˘ t mu.c tieˆu nghieˆn c´ u.u veˆ` t´ınh lieˆn thoˆng v`a su toˆ`n ta.i chu tr`ınh Hamilton c´ac d¯oˆ` thi meta luaˆn ho`an baˆc Keˆ´t qua’ cu’a luaˆn a´n ch´ınh l`a vieˆc d¯a˘ c tru ng t´ınh lieˆn thoˆng cu’a d¯oˆ` thi meta luaˆn ho`an baˆc k˜y thuaˆt du a treˆn moˆt d¯u o c xaˆy du ng cho c´ac d¯oˆ` thi meta luaˆn ho`an toˆ’ng qu´at T` u d¯o´, su toˆ`n ta.i chu tr`ınh Hamilton l´o.p d¯oˆ` thi n`ay d¯a˜ d¯u.o c xem x´et v`a kh˘a’ ng d¯.inh d¯oˆ´i v´o.i moˆt soˆ´ tru.`o.ng ho p Noˆi dung cu’a luaˆn ´an bao goˆ`m phaˆ`n mo’ d¯aˆ`u, phaˆ`n keˆ´t luaˆn v`a ba chu.o.ng: u.c co ba’n; Chu.o.ng C´ac kieˆ´n th´ Chu.o.ng T´ınh lieˆn thoˆng cu’a d¯oˆ` thi meta luaˆn ho`an baˆc 4; Chu.o.ng Chu tr`ınh Hamilton d¯oˆ` thi meta luaˆn ho`an baˆc Chu.o.ng tr`ınh b`ay v˘a´n t˘a´t nh˜ u.ng kh´ai nieˆm co ba’n cu’a l´y thuyeˆ´t d¯oˆ` thi., l´y thuyeˆ´t nh´om ho´an vi v`a moˆt soˆ´ vaˆ´n d¯ˆe` lieˆn quan d¯ˆe´n d¯oˆ´i tu.o ng nghieˆn c´ u.u cu’a luaˆn ´an l`a d¯oˆ` thi meta luaˆn ho`an Chu.o.ng tr`ınh b`ay c´ac keˆ´t qua’ veˆ` t´ınh lieˆn thoˆng cu’a d¯oˆ` thi meta luaˆn ho`an baˆc C´ac d¯.inh l´y 2.5, 2.11 l`a d¯ieˆ`u kieˆn caˆ`n v`a d¯u’ d¯ˆe’ moˆt d¯oˆ` - ˆe’ ch´ thi meta luaˆn ho`an baˆc u.ng minh c´ac d¯.inh l´y n`ay, lieˆn thoˆng D d¯ˆe` 2.1, 2.2 v`a mu.c 2.1 d¯a˜ d¯u.a k˜y thuaˆt toˆ’ng qu´at c´ac meˆnh 2.3 c` ung v´o.i vieˆc d¯ˆe` 1.1 K˜y thuaˆt ´ap du.ng Meˆnh n`ay c´o theˆ’ ´ap du.ng cho mo.i d¯oˆ` thi meta luaˆn ho`an neˆn c˜ ung c´o gi´a tri d¯oˆc laˆp nhaˆ´t d¯.inh Chu.o.ng d¯ˆe` caˆp t´o i su toˆ`n ta.i chu tr`ınh Hamilton c´ac d¯oˆ` thi meta luaˆn ho`an baˆc soˆ´ d¯ieˆ`u lieˆn thoˆng Keˆ´t qua’ ch´ınh o’ d¯ˆay l`a moˆt - oˆ´i v´o.i c´ac d¯oˆ` kieˆn d¯u’ d¯ˆe’ c´ac d¯oˆ` thi d¯ang x´et c´o chu tr`ınh Hamilton D Footer Page of 123 Header Page 10 of 123 10 u nhaˆ´t kh´ac roˆ˜ng, c´ac d¯.inh l´y 3.6, 3.7, 3.8 v`a 3.9 d¯a˜ thi c´o bieˆ’u tu.o ng th´ kh˘a’ ng d¯.inh su toˆ`n ta.i cu’a chu tr`ınh Hamilton moˆt soˆ´ tru `o ng ho p - i.nh l´y 3.10 c˜ u nhaˆ´t cu’a c´ac d¯oˆ` thi n`ay l`a roˆ˜ng, D ung Khi bieˆ’u tu.o ng th´ chı’ d¯u.o c moˆt ung c´o chu tr`ınh Hamilton neˆ´u v`ai d¯ieˆ`u kieˆn d¯u’ d¯ˆe’ ch´ m = C´ac keˆ´t qua’ d¯a˜ d¯o´ng g´op phaˆ`n n`ao v`ao vieˆc l`am s´ang to’ theˆm cho gia’ thuyeˆ´t cu’a Thomassen hay gia’ thuyeˆ´t cu’a Alspach v`a Parsons n´oi r˘`a ng: Taˆ´t ca’ c´ac d¯oˆ` thi meta luaˆn ho`an kh´ac v´o.i d¯oˆ` thi Petersen d¯ˆe`u c´o chu tr`ınh Hamilton C´ac keˆ´t qua’ cu’a luaˆn ´an d¯u o c coˆng boˆ´ c´ac b`ai b´ao [39], [40], [41] v`a d¯a˜ d¯u.o c b´ao c´ao ta.i: • Seminar “Co so’ To´an ho.c cu’a Tin ho.c”, Vieˆn To´an ho.c, Vieˆn Khoa ho.c v`a Coˆng ngheˆ Vieˆt Nam, H`a Noˆi; • Hoˆi nghi Quoˆ´c teˆ´ “Co so’ To´an ho.c cu’a Tin ho.c” (MFI 99), 10/1999, H`a Noˆi; ´ ng du.ng”, 12/2001, H`a Noˆi; • Hoˆi nghi Quoˆ´c teˆ´ “Toˆ’ ho p v`a U • Hoˆi u 6, 09/2002, Hueˆ´; nghi To´an ho.c To`an quoˆ´c laˆ`n th´ • Tru.`o.ng thu “Co so’ To´an ho.c cu’a Tin ho.c”, 09/2003, Qui Nho.n Luaˆn To´an ho.c, Vieˆn ´an d¯u o c ho`an th`anh ta.i Vieˆn Khoa ho.c v`a Coˆng ngheˆ Vieˆt Nam, du ´o i su hu ´o ng daˆ˜n khoa ho.c cu’a PGS TS Ngoˆ - `ao - ˘a´c Taˆn, Vieˆn -u D ´.c Th`anh, Boˆ Gi´ao du.c v`a D To´an ho.c v`a TS Kieˆ`u D ta.o Toˆi xin b`ay to’ l`ong bieˆ´t o.n chaˆn th`anh v`a saˆu s˘a´c t´o.i c´ac thaˆ`y u.ng ngu.`o.i d¯a˜ ta.o toˆi nieˆ`m say meˆ khoa ho.c, hu.´o.ng daˆ˜n, nh˜ tinh thaˆ`n l`am vieˆc uc v`a d¯a˜ d`anh cho toˆi su hu.´o.ng daˆ˜n chı’ nghieˆm t´ ung qu´ı b´au Rieˆng v´o.i thaˆ`y Kieˆ`u ba’o c´o d¯oˆi ch´ ut kh˘a´t khe nhu.ng voˆ c` -u D ´.c Th`anh, toˆi muoˆ´n d¯u.o c b`ay to’ nieˆ`m thu.o.ng tieˆ´c chaˆn th`anh Moˆt tai na.n chuyeˆ´n coˆng t´ac d¯a˜ cu ´o p d¯i sinh ma.ng cu’a thaˆ`y, ngu `o.i u.ng bu.´o.c toˆi m´o.i chaˆp u.ng bu.´o.c v`ao d¯a˜ d`ıu d˘a´t toˆi t` u.ng bu.´o.c, t` ch˜ d¯u.`o.ng nghieˆn c´ u.u To´an ho.c Footer Page 10 of 123 Header Page 73 of 123 73 u vi0 t´o.i vj1 ,o’ Hamilton T` u d¯o´ suy G11 c´o moˆt d¯u `o ng Hamilton P t` d¯ˆay j − i ∈ S1 Khi d¯o´ ψ(P ) c˜ ung l`a moˆt u d¯u `o ng Hamilton G22 t` 0 ψ(vi0) = vi+ o.i ψ(vj1 ) = vj+ n t´ n Ta la i thaˆ´y, G vi keˆ` v´o i vi+ n2 c`on 2 ˆn ta c´o theˆ’ xaˆy du ng d¯u.o c moˆt vj1 keˆ` v´o.i vj+ n , ne chu tr`ınh Hamilton 1 ung v´o.i c´ac ca.nh vi0 vi+ n, v v G t` u c´ac d¯u.`o.ng P , ψ(P ) c` j j+ n2 - ˘a.t G12 = G[V12] Baˆy gi`o ta la.i gia’ thieˆ´t r˘a` ng ca’ h, k v`a d¯ˆe`u le’ D v`a G21 = G[V21] X´et c´ac d¯oˆ` thi G12, G21 theo c´ach tu.o.ng tu nhu treˆn, ta c˜ ung chı’ d¯u.o c r˘a` ng d¯oˆ` thi G c´o chu tr`ınh Hamilton T´o.i d¯ˆay, ph´ep ch´ u.ng minh Boˆ’ d¯ˆe` 3.5 d¯u.o c ho`an thieˆn 3.2 - ie ˆn D ¯u’ cho su to ˆ`u kie ˆ`n ta.i chu tr`ınh Hamilton d Trong mu.c n`ay, ta ch´ u.ng minh su toˆ`n ta.i chu tr`ınh Hamilton moˆt soˆ´ d¯oˆ` thi meta luaˆn ho`an baˆc - i.nh l´ a d¯oˆ` thi meta D y 3.6 Gia’ su’ G = MC(m, n, α, S0, S1 , , Sµ ) l` ˆn ho` lua an baˆc o.i S0 = ∅ v`a m = ho˘a.c m = Khi d¯´o G lieˆn thoˆng v´ c´o chu tr`ınh Hamilton Ch´ u.ng minh Gia’ su’ G = MC(m, n, α, S0, S1, , Sµ) l`a moˆt d¯oˆ` thi meta luaˆn ho`an baˆc lieˆn thoˆng v´o i S0 = ∅ v`a m = ho˘a.c m = - i.nh l´y 2.11, chı’ moˆt Theo D c´ac tru `o ng ho p sau c´o theˆ’ xa’y ra: m = 1, S0 = {±s, ±r} v`a gcd(s, r, n) = 1; m = 2, n ch˘a˜n, S0 = {±s, n2 }, S1 = {k} v`a gcd(s, n2 ) = 1; m = 2, S0 = {±s}, S1 = {k, } v`a gcd(s, k − , n) = 1; m = 2, n ch˘a˜n, S0 = { n2 }, S1 = {h, k, } v`a gcd(h − k, k − , n2 ) = - i.nh l´y 3.6 d¯u.o c suy t` V`ı vaˆy u c´ac boˆ’ d¯ˆe` 3.1, 3.3, 3.4 v`a 3.5 D Footer Page 73 of 123 Header Page 74 of 123 74 Tieˆ´p tu.c xem x´et c´ac d¯oˆ` thi meta luaˆn ho`an baˆc ung lieˆn thoˆng c˜ u nhaˆ´t kh´ac roˆ˜ng nhu.ng soˆ´ khoˆ´i m > 2, ch´ ung ta ch´ u.ng c´o bieˆ’u tu.o ng th´ minh d¯u.o c c´ac d¯.inh l´y sau - inh l´ D y 3.7 Gia’ su’ G = MC(m, n, α, S0, S1 , , Sµ ) l` a d¯oˆ` thi meta ˆn ho` lua an baˆc lieˆn thoˆng v´o i S0 = ∅, m > v`a ca’ m v`a n d¯ˆe`u le’ Khi d¯´o G c´ o chu tr`ınh Hamilton Ch´ u.ng minh X´et d¯oˆ` thi G = MC(m, n, α, S0, S1, , Sµ) tho’a m˜an - i.nh l´y 2.11, ta pha’i d¯ieˆ`u kieˆn cu’a d¯.inh l´y Khi d¯o´ m ≥ Theo D c´o S0 = {±s}, Si = {k} v´o.i i n`ao d¯o´ thuoˆc {1, 2, , µ} cho gcd(i, m) = 1, Sj = ∅ v´o.i mo.i i = j ∈ {1, 2, , µ} v`a gcd(s, r, n) = o’ d¯ˆay r = k(1 + αi + α2i + · · · + α(m−1)i ) Gia’ su’ G = MC(m, n, α , S0, S1, , Sµ ) l`a d¯oˆ` thi meta luaˆn ho`an v´o.i V (G ) = uxy | x ∈ Zm, y ∈ Zn v`a α = αi , S0 = S0 , S1 = Si , S2 = S3 = · · · = Sµ = ∅ X´et ´anh xa ϕ : V (G) → V (G ), vyxi → uxy Do gcd(i, m) = 1, ta c´o u.a, neˆ´u vyxi vhxi+r ∈ E(G) theˆ’ thaˆ´y r˘a` ng ϕ l`a moˆt song a´nh Ho n theˆ´ n˜ th`ı pha’i c´o ho˘a.c r = i v`a (h − y) ∈ αxi Si ho˘a.c r = v`a (h − y) ∈ αxi S0 Neˆ´u r = i v`a (h − y) ∈ αxi Si th`ı ϕ(vyxi)ϕ(vhxi+i) = uxyux+1 v´o.i h - ieˆ`u n`ay c´o ngh˜ıa l`a (h − y) ∈ (αi )xSi , t´ (h − y) ∈ αxi Si D u.c l`a (h − y) ∈ x x+1 xi (α )xS1 Vaˆy uy uh l`a moˆt ca.nh cu’a G Neˆ´u r = v`a (h − y) ∈ α S0 th`ı ϕ(vyxi)ϕ(vhxi+0) = uxyuxh v´o.i (h − y) ∈ αxi S0 = (α )xS0 V`a v`ı theˆ´ uxy uxh c˜ ung l`a moˆt ca.nh cu’a G Ho`an to`an tu o ng tu , ta c´o theˆ’ kieˆ’m tra d¯u o c −1 x −1 x+r r˘a` ng neˆ´u uxy uhx+r l`a moˆt ung l`a moˆt ca.nh cu’a G th`ı ϕ (uy )ϕ (uy ) c˜ ca.nh cu’a G Vaˆy, u G leˆn G V`ı vaˆy, khoˆng l`am maˆ´t ϕ l`a moˆt d¯a˘’ ng caˆ´u t` t´ınh toˆ’ng qu´at, ta c´o theˆ’ gia’ thieˆ´t r˘a` ng, G = MC(m, n, α, S0, S1, , Sµ ) v´o.i m > le’ , n le’ , S0 = {±s}, S1 = {k}, S2 = S3 = · · · = Sµ = ∅ v`a gcd(s, r, n) = 1, o’ d¯ˆay r = k(1 + α + α2 + · · · + α(m−1) ) Footer Page 74 of 123 Header Page 75 of 123 75 i Gia’ su’ ρ l`a tu d¯a˘’ ng caˆ´u cu’a G x´ac d¯.inh bo’.i ρ(vji ) = vj+1 Khi d¯o´ ρ l`a nu’.a ch´ınh qui Neˆ´u gcd(s, n) = d th`ı tu d¯a˘’ ng caˆ´u β = ρd u.a d¯ı’nh vji ch´ınh l`a Vji = c˜ ung l`a nu’.a ch´ınh qui Qu˜ı d¯a.o cu’a β ch´ i i i , vj+2d , , vj+( vji , vj+d n −1)d d n i i M˘a.t kh´ac, c´ac taˆp 0, d, 2d, , ( d − 1)d v`a 0, α s, 2α s, , ung neˆn G[Vji ] ch´ınh l`a chu tr`ınh ( nd − 1)αis cu’a Zn tr` i i i i vji vj+α i s vj+2αi s vj+( n −1)αi s vj d v´o.i i = 0, 1, , (m − 1); j = 0, 1, , (d − 1) i Neˆ´u β c´o caˆ´p th`ı ρ2d (vji ) = vji , t´ u.c l`a vj+2d = vji ⇔ 2d ≡ (mod n) - ieˆ`u n`ay l`a khoˆng theˆ’ v`ı n le’ v`a d l`a moˆt D u ´o c thu c su cu’a n X´et d¯oˆ` thi thu.o.ng G/β Ta c´o V (G/β) = Vji | i ∈ Zm , j ∈ Zd v`a hai d¯ı’nh cu’a G/β (l`a hai qu˜ı d¯a.o cu’a β ) keˆ` G/β neˆ´u c´o moˆt d¯ı’nh thuoˆc qu˜ı d¯a.o n`ay v´o i moˆt d¯ı’nh ca.nh G noˆ´i moˆt thuoˆc ung lieˆn thoˆng qu˜ı d¯a.o Do d¯oˆ` thi G lieˆn thoˆng neˆn G/β c˜ Theˆm n˜ u.a, G[Vji ] l`a moˆt chu tr`ınh v`a G c´o baˆc b˘`a ng neˆn d¯oˆ` thi thu.o.ng G/β l`a ch´ınh qui baˆc Suy G/β l`a moˆt chu tr`ınh Ta la.i ung c´o |V (G/β)| = md v´o.i m le’ v`a d l`a moˆt u ´o c cu’a n neˆn |V (G/β)| c˜ - i.nh l´y 3.7 - i.nh l´y 1.5, ta keˆ´t luaˆn le’ Theo D G c´o chu tr`ınh Hamilton D d¯u.o c ch´ u.ng minh - i.nh l´y 3.7 veˆ` su V´ı du sau d¯ˆay s˜e minh ho.a cho kh˘a’ ng d¯.inh cu’a D toˆ`n ta.i cu’a chu tr`ınh Hamilton c´ac d¯oˆ` thi meta luaˆn ho`an baˆc lieˆn thoˆng c´o bieˆ’u tu.o ng th´ u nhaˆ´t kh´ac roˆ˜ng v`a ca’ m, n d¯ˆe`u le’ V´ı du H`ınh 3.1 l`a moˆt chu tr`ınh Hamilton d¯oˆ` thi G = MC(3, 7, 2, {±1}, {0}) - inh l´ D y 3.8 Gia’ su’ G = MC(m, n, α, S0, S1, , Sµ) l` a d¯oˆ` thi meta ˆn ho` lua an baˆc o m > 2, S0 = {±s}, Si = {k} v´o.i i n`ao d¯´o c´ thuoˆc a˜n v` a {1, 2, , µ} neˆ´u m le’ ho˘a.c thuoˆc {1, 2, , µ − 1} neˆ´u m ch˘ Footer Page 75 of 123 Header Page 76 of 123 76 v00 s v10 s sv0 v11 s s s v0 v21 s s v1 v s v02 v12 v31 s v30 s s v2 s v22 sv60 s s s v42 v32 sv5 s v sv s v40 - inh l´ H`ınh 3.1: V´ı du minh ho.a cho D y 3.7 gcd(i, m) = 1, Sj = ∅ v´ o.i mo.i j ∈ {1, 2, , µ} \ {i} Neˆ´u gcd(r, n) = 1, d¯´o r = k(1 + αi + · · · + α(m−1)i ) th`ı G c´ o chu tr`ınh Hamilton - i.nh l´y Ch´ u.ng minh Tru.´o.c heˆ´t ta thaˆ´y G l`a d¯oˆ` thi lieˆn thoˆng theo D x 2.5 Gia’ su’ G c´o taˆp d¯ı’nh V = {vy | x ∈ Zm ; y ∈ Zn } Tu d¯a˘’ ng caˆ´u x ρ : vyx → vy+1 treˆn G l`a nu’.a ch´ınh qui neˆn ta c´o theˆ’ x´et d¯oˆ` thi thu.o.ng - ˘a.t V x = {vyx| y ∈ Zn }; Gx = G[V x ], x = 0, , m − Khi d¯o´ V x G/ρ D l`a c´ac d¯ı’nh cu’a G/ρ Do Si = {k} neˆn theo d¯.inh ngh˜ıa d¯oˆ` thi thu.o.ng, V xV x+i l`a ca.nh cu’a G/ρ X´et chu tr`ınh C = V V i V 2i V (m−1)iV G/ρ La.i c´o gcd(i, m) = neˆn {0, i, 2i, , (m − 1)i} l`a taˆ´t ca’ c´ac phaˆ`n tu’ cu’a Zm V`ı vaˆy C l`a chu tr`ınh Hamilton G/ρ Ta xaˆy i u C nhu sau: P xuaˆ´t ph´at t` u vy0 cu’a G0 , d¯i t´o.i vy+k cu’a du ng d¯u.`o.ng P t` (m−1)i 2i ’ a G2i Tieˆ´p tu.c nhu vaˆy, Gi , roˆ`i t´o.i vy+k(1+α i ) cu t´o i vy+k(1+αi+···+α(m−2)i ) cu’a G(m−1)i v`a quay tro’ veˆ` vy+r cu’a G0 Taˆp taˆ´t ca’ c´ac d¯u `o ng d¯i d¯u o c xaˆy du ng theo c´ach th´ u.c treˆn, [2], d¯u.o c k´y hieˆu l`a coil(C) Trong d¯oˆ` thi G, tu d¯a˘’ ng caˆ´u ρ nu’.a ch´ınh qui v`a c´o caˆ´p n, d¯oˆ` thi u.a c´ac d¯u.`o.ng P thu.o.ng G/ρ c´o chu tr`ınh Hamilton C v`a coil(C) ch´ noˆ´i hai d¯ı’nh cu’a G0 c´o khoa’ng c´ach r = k(1 + αi + · · · + α(m−1)i) v´o.i gcd(r, n) = Theo Boˆ’ d¯ˆe` [26], G c´o chu tr`ınh Hamilton Footer Page 76 of 123 Header Page 77 of 123 77 - i.nh l´ D y 3.9 Gia’ su’ G = MC(m, n, α, S0, S1, , Sµ) l` a d¯oˆ` thi meta ˆn ho` lua an baˆc o m chia heˆ´t cho 4, n ch˘a˜n, S0 = { n2 }, lieˆn thoˆng c´ ao d¯´ o thuoˆc o.i i n` Si = {s} v´ {1, 2, , µ − 1} cho gcd(i, m) = 1, Sj = ∅ cho mo.i j ∈ {1, 2, , µ − 1} \ {i}, Sµ = {r} Khi d¯´o G c´o chu tr`ınh Hamilton x Ch´ u.ng minh Gia’ su’ G c´o taˆp d¯ı’nh V = {vy | x ∈ Zm ; y ∈ Zn } K´y x x x hieˆu wy = {vy , vy+ n2 }, o’ d¯ˆay x ∈ Zm, y ∈ Zn/2 X´et d¯oˆ` thi G d¯u o c xaˆy x u G nhu sau: G c´o taˆp du ng t` d¯ı’nh V (G ) = {wy | x ∈ Zm , y ∈ Zn/2}, hai d¯ı’nh wyx v`a whk , k = x, l`a keˆ` G v`a chı’ toˆ`n ta.i u ∈ wyx v`a v ∈ whk cho u, v keˆ` G M˘a.t kh´ac, deˆ˜ kieˆ’m tra d¯u.o c G d¯a˘’ ng caˆ´u v´o.i d¯oˆ` thi meta luaˆn ho`an MC(m, n2 , α , S0, , Sµ ), d¯o´ α ≡ α (mod n2 ), Si = {s } v´o.i s ≡ s (mod n2 ), Sj = ∅ v´o.i mo.i j ∈ {0, 1, , µ − 1} \ {i} c`on Sµ = {r } v´o.i r ≡ r (mod n2 ) Do vaˆy ta c´o theˆ’ d¯oˆ`ng nhaˆ´t G v´o i d¯oˆ` thi n`ay Nhu u.a, G vaˆy d¯oˆ` thi meta luaˆn ho`an baˆc G l`a moˆt v´o i S0 = ∅ Theˆm n˜ - i.nh l´y [29], G s˜e c´o chu l`a lieˆn thoˆng v`a c´o m chia heˆ´t cho Theo D tr`ınh Hamilton Tieˆ´p theo ta s˜e xaˆy du ng chu tr`ınh Hamilton C G t` u moˆt chu tr`ınh Hamilton C G b˘`a ng c´ach sau d¯ˆay xt−1 Gia’ su’ C = w00 wyx11 wyx22 wyt−1 w0 l`a moˆt chu tr`ınh Hamilton c´o d¯oˆ u d`ai t G Tru.´o.c heˆ´t ta xaˆy du ng d¯u.`o.ng P1 G nhu sau: T` x d¯ı’nh z00 = v00 ∈ w00 ta d¯i t´o.i d¯ı’nh zyx11 thuoˆc wy11 b˘`a ng moˆt ca.nh x ung b˘a` ng moˆt G Tieˆ´p theo, t` u zyx11 ta d¯i t´o.i d¯ı’nh zyx22 thuoˆc wy22 c˜ ca.nh xt−1 xt−1 d¯a˜ d¯u.o c cho.n thuoˆc G, , v`a cuoˆ´i c` ung t` u d¯ı’nh zyt−1 wyt−1 , b˘`a ng x moˆt ca.nh G, ta d¯i t´o i d¯ı’nh zytt cu’a w0 C´ach xaˆy du ng d¯u `o ng P1 nhu treˆn l`a thu c hieˆn d¯u o c d¯.inh ngh˜ıa cu’a d¯oˆ` thi G Nhu vaˆy ta c´o t−1 xt P1 = z00 zyx11 zyx22 zyxt−1 zyt , u P1 b˘`a ng c´ach thay o’ d¯ˆay z00 = v00 Baˆy gi`o ta la.i xaˆy du ng d¯u.`o.ng P2 t` Footer Page 77 of 123 Header Page 78 of 123 78 theˆ´ moˆ˜i d¯ı’nh zyxii P1 bo’.i zyxii+ n C´o hai kha’ n˘ang xa’y ra: Kha’ n˘ ang 1: zyxtt = z 0n Trong tru.`o.ng ho p n`ay, zyxtt+ n = z00 V`ı vaˆy, C = P1 ∪ P2 l`a chu tr`ınh Hamilton G Kha’ n˘ ang 2: zyxtt = z00 L´ uc n`ay, zyxtt+ n = z 0n v`a v`ı theˆ´, ca’ P1 v`a P2 d¯ˆe`u l`a chu tr`ınh 2 xt−1 xt−1 n ` ˆy keˆ` v´o.i zyt−1 G Do S0 = { } neˆn z0 keˆ v´o.i z 0n v`a zyt−1 + n Do va 2 0 t−1 t−2 t−1 C = z00 zyx11 zyx22 zyxt−1 zyt−1 + n zyt−2 + n z n2 z0 x x 2 u.ng minh l`a chu tr`ınh Hamilton G T´o.i d¯ˆay, d¯.inh l´y d¯u.o c ch´ - i.nh l´y 3.9 veˆ` c´ac d¯oˆ` thi meta luaˆn ho`an baˆc D lieˆn thoˆng c´o m chia heˆ´t cho v`a S0 = ∅ s˜e d¯u.o c minh ho.a v´ı du sau d¯ˆay V´ı du H`ınh 3.2 l`a moˆt chu tr`ınh Hamilton G = MC(4, 6, 1, {3}, {1}, {0}) d¯a˜ x´oa d¯i c´ac ca.nh cu’a G khoˆng thuoˆc chu tr`ınh n`ay tv4 v53 t tv4 v52 t vt41 v51 t v50 t vt03 v0t vt30 t t t t t vt 00 tv01 v13 v40 v12 v11 v10 vt31 vt32 vt33 t v20 t v21 t v22 t3 v2 - i.nh l´ H`ınh 3.2: V´ı du minh ho.a cho D y 3.9 Footer Page 78 of 123 Header Page 79 of 123 79 ung ta d¯a˜ ch´ u.ng minh d¯u.o c su toˆ`n ta.i chu O’ c´ac d¯.inh l´y treˆn, ch´ tr`ınh Hamilton moˆt soˆ´ d¯oˆ` thi meta luaˆn ho`an baˆc c´o bieˆ’u tu o ng u nhaˆ´t cu’a ch´ ung b˘a` ng roˆ˜ng, keˆ´t th´ u nhaˆ´t kh´ac roˆ˜ng Khi bieˆ’u tu.o ng th´ qua’ sau d¯ˆay l`a d¯ieˆ`u kieˆn d¯u’ cho tru `o ng ho p d¯oˆ` thi d¯ang x´et c´o khoˆ´i - i.nh l´ D y 3.10 Gia’ su’ G = MC(2, n, α, S0, S1 ) l`a d¯oˆ` thi meta luaˆn ho` an baˆc o S0 = ∅, S1 = {r1, r2, r3, r4} Khi d¯´o G c´o chu tr`ınh c´ Hamilton neˆ´u tho’ a m˜ an moˆt hai d¯ieˆ`u kieˆn sau: ao d¯´ o thuoˆc Toˆ`n ta.i i n` {1, 2, 3} cho gcd(ri − r4, n) = 1; Toˆ`n ta.i j, k ∈ {1, 2, 3}, j = k cho gcd(rj − r4 , rk − r4, n) = x Ch´ u.ng minh Gia’ su’ d¯oˆ` thi G c´o taˆp d¯ı’nh V = {vy | x ∈ Z2 ; y ∈ Zn} x X´et d¯oˆ` thi G = MC(2, n, −1, S0, S1) c´o taˆp d¯ı’nh V (G ) = {wy | x ∈ Z2, y ∈ Zn } v`a S1 = {r1 − r4, r2 − r4 , r3 − r4, r4 − r4 } C´o theˆ’ kieˆ’m tra l`a d¯u.o c song a´nh ϕ : V (G) → V (G ) x´ac d¯.inh bo’.i vy0 → wy0, vy1 → wy−r moˆt u.a G v`a G d¯a˘’ ng caˆ´u gi˜ Do vaˆy, khoˆng l`am maˆ´t t´ınh toˆ’ng qu´at, ta c´o theˆ’ gia’ thieˆ´t d¯oˆ` thi G = MC(2, n, α, S0, S1) d¯a˜ cho c´o α = −1 v`a S1 = {r1, r2, r3, 0} Nhu u.ng minh G c´o chu tr`ınh vaˆy, u.ng minh meˆnh d¯ˆe` n`ay, ta chı’ caˆ`n ch´ d¯ˆe’ ch´ Hamilton neˆ´u n´o tho’a m˜an moˆt c´ac d¯ieˆ`u kieˆn: Toˆ`n ta.i i n`ao d¯o´ thuoˆc {1, 2, 3} cho gcd(ri, n) = 1; Toˆ`n ta.i j, k ∈ {1, 2, 3}, j = k cho gcd(rj , rk , n) = Tru.´o.c heˆ´t ca’ hai tru.`o.ng ho p treˆn, d¯oˆ` thi G l`a lieˆn thoˆng theo - i.nh l´y 2.11 Baˆy gi`o ta x´et t` u.ng tru.`o.ng ho p D Toˆ`n ta.i i n` ao d¯´ o thuoˆc {1, 2, 3} cho gcd(ri, n) = 1 Trong G x´et chu tr`ınh C c´o da.ng: C = v00 vr1i vr0i v2r v v(n−1)r v1v0 i 2ri i 0 Do gcd(ri, n) = neˆn {0, ri, 2ri, , (n − 1)ri} l`a taˆ´t ca’ c´ac phaˆ`n tu’ cu’a Zn Vaˆy C l`a moˆt chu tr`ınh Hamilton G Footer Page 79 of 123 Header Page 80 of 123 80 Toˆ`n ta.i j, k ∈ {1, 2, 3}, j = k cho gcd(rj , rk , n) = X´et d¯oˆ` thi G cu’a G x´ac d¯.inh bo’.i G = MC(2, n, −1, S0, S1) v´o.i um cu’a G, c´o baˆc S1 = {rj , rk , 0} Hieˆ’n nhieˆn, G l`a d¯oˆ` thi bao tr` G - i.nh l´y [27]) La.i theo D - i.nh lieˆn thoˆng c´o gcd(rj , rk , n) = (theo D l´y [5], G l`a d¯oˆ` thi Cayley treˆn ρ, τ v´o.i ρ, τ l`a c´ac tu d¯a˘’ ng caˆ´u cu’a G x+1 x x+1 x´ac d¯.inh bo’.i ρ(vyx ) = vy+1 v`a τ (vyx) = vαy = v−y M˘a.t kh´ac, deˆ˜ kieˆ’m u.c ρn = τ = v`a τ ρτ −1 = ρ−1 tra d¯u.o c r˘`a ng ρ, τ tho’a m˜an c´ac heˆ th´ neˆn ρ, τ l`a moˆt nh´om nhi dieˆn Nhu vaˆy, G l`a d¯oˆ` thi Cayley baˆc lieˆn thoˆng treˆn nh´om nhi dieˆn - i.nh l´y 1.4, G c´o chu tr`ınh Hamilton Do G l`a d¯oˆ` thi ρ, τ Theo D bao tr` um cu’a G neˆn c´o theˆ’ keˆ´t luaˆn ung c´o chu tr`ınh Hamilton G c˜ - i.nh l´y 3.10 s˜e d¯u.o c moˆ ta’ b˘`a ng moˆt Keˆ´t qua’ cu’a D chu tr`ınh Hamilton d¯oˆ` thi G = MC(2, 7, 1, ∅, {0, 2, 3, 5}) o’ v´ı du sau d¯ˆay V´ı du Trong h`ınh 3.3 l`a moˆt chu tr`ınh Hamilton d¯oˆ` thi meta luaˆn ho`an baˆc u nhaˆ´t l`a taˆp c´o m = v`a bieˆ’u tu o ng th´ roˆ˜ng v00 v10 v20 v30 v40 v50 v60 v01 v11 v21 v31 v41 v51 v61 t ❏ ❏ t ❏ t ❏ ❏ ❏ t ❏ ❏ t ❏ t t ✧ ✧ ✧ ✧ ✧ ✧ ✧ ❏✧ ❏ ❏ ✧✧ ❏ ✧✧ ❏ ❏ ✧ ✧ ❏ ❏ ❏ ✧❏ ✧❏ ❏ ✧ ❏ ✧ ❏ ❏ ❏ ✧ ✧ ❏ ❏ ❏ ✧❏ ✧❏ ❏ ❏ ❏ ✧✧ ❏ ✧✧ ❏ ✧❏ ❏ ✧❏ ❏ ❏ ❏ ❏ ❏ ❏✧✧ ❏✧✧ ✧❏ ✧❏ ❏ ❏ ❏ ✧ ✧ ✧ ❏ ❏ ❏ ❏ ❏✧ ✧ ✧ ❏ ❏ ❏ ❏ ✧ ✧ ❏ ❏t ✧ ❏t ❏t ❏t ❏t t✧ t - i.nh l´ H`ınh 3.3: V´ı du minh ho.a cho D y 3.10 ung ta d¯a˜ chı’ d¯u.o c d¯ieˆ`u kieˆn T´om la.i, o’ chu.o.ng n`ay ch´ d¯u’ cho su toˆ`n ta.i chu tr`ınh Hamilton moˆt soˆ´ d¯oˆ` thi meta luaˆn ho`an baˆc u u treˆn c´ac C´ac d¯.inh l´y 3.6, 3.7, 3.8 v`a 3.9 l`a nh˜ u ng keˆ´t qua’ nghieˆn c´ Footer Page 80 of 123 Header Page 81 of 123 81 u nhaˆ´t kh´ac d¯oˆ` thi meta luaˆn ho`an baˆc lieˆn thoˆng c´o bieˆ’u tu o ng th´ - i.nh l´y 3.10 d¯a˜ x´et d¯ˆe´n c´ac d¯oˆ` thi n`ay ch´ roˆ˜ng D ung c´o bieˆ’u tu.o ng ung ta d¯a˜ su’ du.ng t´o.i moˆt th´ u nhaˆ´t b˘a` ng roˆ˜ng O’ d¯ˆay, ch´ v`ai keˆ´t qua’ tru.´o.c d¯ˆay veˆ` d¯oˆ` thi Cayley, d¯oˆ` thi thu.o.ng, d¯oˆ` thi Petersen toˆ’ng qu´at GP (n, k) v`a c´ac k˜y thuaˆt cu’a toˆ’ ho p d¯ˆe’ xaˆy du ng tru c tieˆ´p chu tr`ınh Hamilton d¯oˆ` thi Tieˆ´p tu.c mo’ roˆng u.u theo hu.´o.ng n`ay, hy vo.ng r˘`a ng nghieˆn c´ ung ta s˜e c´o theˆm nh˜ u.ng keˆ´t qua’ saˆu s˘a´c ho.n treˆn l´o.p th`o.i gian t´o.i, ch´ d¯oˆ` thi d¯ang x´et Footer Page 81 of 123 Header Page 82 of 123 ˆN ˆ´T LUA KE Taˆ´t ca’ c´ac keˆ´t qua’ d¯u.o c tr`ınh b`ay luaˆn ´an d¯ˆe`u xoay quanh l´o.p d¯oˆ` thi meta luaˆn ho`an baˆc 4, d¯oˆ´i tu o ng ch´ınh luaˆn ´an V´o i hai mu.c d¯´ıch d¯a˘ t l`a x´et t´ınh lieˆn thoˆng v`a su toˆ`n ta.i chu tr`ınh Hamilton cu’a c´ac d¯oˆ` thi n`ay, luaˆn ´an d¯a˜ d¯a.t d¯u o c c´ac keˆ´t qua’ sau d¯ˆay: Xaˆy du ng k˜y thuaˆt toˆ’ng qu´at d¯ˆe’ x´ac d¯.inh d¯ieˆ`u kieˆn lieˆn thoˆng cho c´ac d¯oˆ` thi meta luaˆn ho`an n´oi chung K˜ y thuaˆt n`ay d¯u o c theˆ’ hieˆn ung v´o.i vieˆc d¯ˆe` 1.1 d¯ˆe` 2.1, 2.2 v`a 2.3 c` c´ac meˆnh ´ap du.ng Meˆnh Su’ du.ng k˜y thuaˆt n´oi treˆn, luaˆn ´an d¯a˜ chı’ d¯u o c d¯ieˆ`u kieˆn caˆ`n - i.nh l´y 2.5 v`a d¯u’ cho t´ınh lieˆn thoˆng cu’a d¯oˆ` thi meta luaˆn ho`an baˆc D u nhaˆ´t kh´ac roˆ˜ng l`a c´ac d¯ieˆ`u kieˆn d`anh cho c´ac d¯oˆ` thi c´o bieˆ’u tu o ng th´ - inh l´y Khi c´ac d¯oˆ` thi c´o bieˆ’u tu.o ng th´ u nhaˆ´t b˘`a ng roˆ˜ng, ch´ ung ta c´o D 2.11 Ngo`ai ra, moˆt ung du a v`ao thu’ tu.c kieˆ’m tra t´ınh lieˆn thoˆng cu’a ch´ hai d¯.inh l´y treˆn c˜ ung d¯u.o c d¯ˆe` xuaˆ´t Veˆ` vaˆ´n d¯ˆe` Hamilton, luaˆn u.ng minh d¯u.o c c´ac d¯.inh l´y ´an d¯a˜ ch´ 3.6, 3.7, 3.8 v`a 3.9 veˆ` d¯ieˆ`u kieˆn d¯u’ cho su toˆ`n ta.i chu tr`ınh Hamilton u nhaˆ´t kh´ac c´ac d¯oˆ` thi meta luaˆn ho`an baˆc c´o bieˆ’u tu o ng th´ - i.nh l´y 3.10 d¯a˜ roˆ˜ng Khi bieˆ’u tu.o ng th´ u nhaˆ´t cu’a ch´ ung b˘a` ng roˆ˜ng, D chı’ d¯u.o c moˆt v`ai d¯ieˆ`u kieˆn d¯ˆe’ d¯oˆ` thi MC(2, n, α, S0, S1 ) v´o i |S1 | = c´o chu tr`ınh Hamilton Tieˆ´p tu.c nghieˆn c´ u.u veˆ` chu tr`ınh Hamilton d¯oˆ` thi meta luaˆn ho`an baˆc ung toˆi d¯ang xem x´et t´o.i c´ac d¯oˆ` thi c´o m chia heˆ´t cho 4, ch´ u.ng keˆ´t qua’ ban d¯aˆ`u Hy vo.ng r˘a` ng th`o.i gian v`a d¯a˜ thu d¯u.o c nh˜ ung toˆi s˜e gia’i quyeˆ´t d¯u.o c tro.n ve.n tru.`o.ng ho p n`ay t´o.i, ch´ 82 Footer Page 82 of 123 Header Page 83 of 123 ´ ˆ ` DANH MU C CAC CONG TRINH ˆng tr`ınh d ˆng bo ˆn quan d ˆn C´ ac co ¯˜ a co o lie ¯e an: ˆ´ c´ ˆ´n lua ´ Ngo Dac Tan and Tran Minh Tuoc, “Connectedness of tetravalent metacirculant graphs with non-empty first symbol”, In: Proceedings of The International Conference “Mathematical Foundation of Informatics” (October 25 – 28, 1999, Hanoi, Vietnam), World Scientific, Singapore (nhaˆn d¯a˘ng) Ngo Dac Tan and Tran Minh Tuoc, “On Hamilton cycles in connected tetravalent metacirculant graphs with non-empty first symbol”, Acta Mathematica Vietnamica 28 (2003), 267 - 278 Ngo Dac Tan and Tran Minh Tuoc, “Connectedness of tetravalent metacirculant graphs with the empty first symbol”, Preprint 2002/33, Institute of Mathematics (2002) (gu’.i d¯a˘ng) ´t b´ ˆi C´ ac t´ om t˘ a ao c´ ac ta.i c´ ac ho nghi.: Ngo Dac Tan and Tran Minh Tuoc, “Connectedness of tetravalent metacirculant graphs with non-empty first symbol”, Abstract of The International Conference “Mathematical Foundation of Informatics”, October 25 – 28, 1999, Hanoi, Vietnam Ngo Dac Tan and Tran Minh Tuoc, “On Hamilton cycles in connected tetravalent metacirculant graphs with non-empty first symbol”, Abstract of The International Conference “Combinatorics and Applications”, December 03 – 05, 2001, Hanoi, Vietnam - ˘a´c Taˆn v`a Traˆ`n Minh Tu.´o.c, “Connectedness of tetravalent Ngoˆ D metacirculant graphs with the empty first symbol”, T´om t˘ a´t c´ac b´ ao c´ao Hoˆi an ho.c To`an quoˆ´c laˆ`n th´ u 6, 07 – 10/09/2002, nghi To´ Hueˆ´, Vieˆt Nam 83 Footer Page 83 of 123 Header Page 84 of 123 ` LIE ˆ U THAM KHA’O TAI [1] Alspach B (1983), “The classification of hamiltonian generalized Petersen graphs”, J Combin Theory Ser B 34, 293 - 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54 Footer Page 85 of 123 Header Page 86 of 123 86 [23] Maruˇsiˇc D and Scapellato R (1994), “Classifying vertex-transitive graphs whose order is a product of two primes”, Combinatorica 14, 184 - 201 [24] Ore O (1960), “A note on Hamiltonian circuits”, Am Math Month 67, 55 [25] Ngo Dac Tan (1990), “On cubic metacirculant graphs”, Acta Mathematica Vietnamica Vol 15, No 2, 57 - 71 [26] Ngo Dac Tan (1992), “Hamilton cycles in cubic (4, n)-metacirculant graphs”, Acta Math Vietnamica Vol 17, No 2, 83 - 93 [27] Ngo Dac Tan (1993), “Connectedness of cubic metacirculant graphs”, Acta Math Vietnamica Vol 18, No.1, - 17 [28] Ngo Dac Tan (1993), “On Hamilton cycles in cubic (m, n)metacirculant graphs”, Australasian Journal of Combinatorics 8, 211 - 232 [29] Ngo Dac Tan (1994), “Hamilton cycles in cubic (m, n)- metacirculant graphs with m divisible by 4”, Graphs and Combin 10, 67 - 73 [30] Ngo Dac Tan (1995), “Hamilton cycles in some vertex-transitive graphs”, SEA Bull Math Vol 19, No 1, 61 - 67 [31] Ngo Dac Tan (1996), “On Hamilton cycles in cubic (m, n)metacirculant graphs, II”, Australasian Journal of Combinatorics 14, 235 - 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232 [39] Ngo Dac Tan and Tran Minh Tuoc, “Connectedness of tetravalent metacirculant graphs with non-empty first symbol”, In: Proceedings of The International Conference “Mathematical Foundation of Informatics” (October 25 – 28, 1999, Hanoi, Vietnam), World Scientific, Singapore (Nhaˆn d¯a˘ng) [40] Ngo Dac Tan and Tran Minh Tuoc (2003), “On Hamilton cycles in connected tetravalent metacirculant graphs with non-empty first symbol”, Acta Mathematica Vietnamica 28, 267 - 278 [41] Ngo Dac Tan and Tran Minh Tuoc, “Connectedness of tetravalent metacirculant graphs with empty first symbol”, Preprint 2002/33, Institute of Mathematics (Gu’.i d¯a˘ng) [42] Wielandt H (1964), Finite permutation groups, Academic Press, New York Footer Page 87 of 123 ... v`a ba chu. o.ng: u.c co ba’n; Chu. o.ng C´ac kieˆ´n th´ Chu. o.ng T´ınh lieˆn thoˆng cu’a d¯oˆ` thi meta luaˆn ho`an baˆc 4; Chu. o.ng Chu tr`ınh Hamilton d¯oˆ` thi meta luaˆn ho`an baˆc Chu. o.ng... d¯u.`o.ng) Hamilton D c´ o chu tr`ınh (t.u d¯u.` o.ng) Hamilton d¯u.o c go.i l`a d¯oˆ` thi Hamilton (t.u nu’.a Hamilton) V´ı du Trong h`ınh 1.11, G l`a d¯oˆ` thi Hamilton, G l`a nu’.a Hamilton. .. du.ng cho mo.i d¯oˆ` thi meta luaˆn ho`an neˆn c˜ ung c´o gi´a tri d¯oˆc laˆp nhaˆ´t d¯.inh Chu. o.ng d¯ˆe` caˆp t´o i su toˆ`n ta.i chu tr`ınh Hamilton c´ac d¯oˆ` thi meta luaˆn ho`an baˆc