1.2. Paths, Cycles, and Trails

Transcription

1 Grph heory 1.2 Ph, Cyle, n Trl 1 emeer Ph, Cyle, n Trl Defnon. Le G e grph. A lk l 0, e 1, 1,, e k, k of ere n ege h h, for 1 k, he ege e h en pon -1 n. A rl lk h no repee ege. A ph grph of G h ph ( ph n e onere lk h no repee ere). A,-lk or,-rl h fr ere n l ere ; hee re enpon. A,-ph ph hoe ere of egree 1( enpon) re n ; he oher re nernl ere. The lengh of lk, rl, ph, or yle nmer of ege. A ngle ere = onere,-lk(rl, ph) of lengh 0. A loop yle of lengh 1. Alo o n ege h he me enpon form yle of lengh 2. A lk or rl loe f enpon re he me. A lk o or een lengh o or een Emple. In he Kongrg grph, he l, e 2,, e 5, y, e 6,, e 1,, e 2, loe lk of lengh 5; repe ege e 2 n hene no rl. e 1 e2 e e 6 5 y e 3 z Whle, e 2,, e 5, y, e 6,, e 1, rl of lengh 4, repe ere no ege. The grph onng of ege e 1, e 5, e 6 n ere,, y yle of lengh 3,; eleng one of ege yel ph. To ege h he me enpon h e 1 n e 2 form yle of lengh Emple. In he mple grph elo, he l,,,,, y,,, y,,,, pefe loe lk of lengh 12, hle,,, y,,, y,,,, pefe loe rl. y Th grph h 5 yle: (,, ), (, y, ), (,, y), (, y, ), (,,, y). The, -rl, y,,, y,, onn he ege of he, -ph, y,,, no of he, -ph, y,. Remrk.1. If e follo ph from o n grph n hen follo ph from o, he rel nee no e, -ph, ee he, -ph n, -ph my he ommon nernl ere, n f he rel, -lk. 2. Syng h lk W onn ph P men h he ere n ege of P or l of he ere n ege of W n orer, neerly onee A lk W : 2, 4, 3, 2, 3, 5, 6, 5 onn ph P : 2, 3, 5, 6. e 4 e 7

2 Grph heory 1.2 Ph, Cyle, n Trl 1 emeer Lemm. In grph G, eery, -lk onn, -ph. Proof: We proe emen y non on he lengh { of, -lk W. B ep: { = 0. Hng no ege, W on of ngle ere (= ). Th ere, -ph of lengh 0. Inon ep: { 1. We ppoe h he lm hol for lk of lengh le hn {. Le W e, -lk of lengh {. If W h no repee ere, hen ere n ege form, -ph. If W h repee ere, hen eleng he ege n ere eeen pperne of (leng one opy of ) yel horer, -lk W onne n W. P By non hypohe, W onn, -ph P, n h ph P onne n W. Eere Coner K 4 : h lk:,,,, h no rl. K 4 h rl:,,,, h no loe n no ph. K 4 h no loe rl h no yle. Sne loe rl h een ere egree, n K 4 h reqre egree 2 or 0, hh o no perm onnee grph h no yle. In f, K 4 h yle of lengh 4:,,,, n h yle of lengh 3:,,, Defnon. A grph G onnee f h, -ph heneer, V(G), ohere, G onnee. If G h, -ph, hen onnee o n G. The onneon relon on V(G) on of he orere pr (, ) h h onnee o. Th, = {(, ) V(G) V(G) : onnee o } relon on V(G). Remrk.1. onnee n jee, e pply only o grph n o pr of ere(e neer y onnee hen ere). 2. The phre onnee o onenen hen rng proof, n opng e m lrfy he non eeen onneon n jeny: G h, -ph n re onnee onnee o E(G) n re jen jone o jen o 3. The onneon relon n eqlene relon on V(G), erfy!. 4. A mml onnee grph of G grph h onnee n no onne n ny oher onnee grph of G Defnon. The omponen of grph G re mml onnee grph,.e. he grph h onnee n no onne n ny oher onnee grph of G. The omponen (or grph) rl f h no ege; ohere nonrl. An ole ere ere of egree 0.

3 Grph heory 1.2 Ph, Cyle, n Trl 1 emeer Remrk. The eqlene le of he onneon relon on V(G) re he ere e of omponen of G. An ole ere form rl omponen, onng of one ere n no ege Emple. The grph elo h 4 omponen, one eng n ole ere. r z y q p The ere e of he omponen re {p}, {q, r}, {,,,, }, n {, y, z}; hee re he eqlene le of he onneon relon. Eere Proe h f ome ere of grph G onnee o ll oher ere of G hen G onnee. Proof. Le n e ny ere n V(G). By he mpon, here e n V(G) h h onnee o n onnee o. Bee, -ph n, -ph ogeher onn, -ph, o onnee o. Then eery ere onnee o eery oher, n G onnee Remrk. Componen re pre jon; no o hre ere. Ang n ege h enpon n n omponen omne hem no one omponen. Th ng n ege eree he nmer of omponen y 0 or 1, n eleng n ege nree he nmer of omponen y 0 or Propoon. Eery grph h n ere n k ege h le n k omponen. Proof: An n-ere grph h no ege h n omponen. By Remrk , eh ege e ree h y mo 1, o hen k ege he een e he nmer of omponen ll le n k. Remrk. 1.When e on grph y eleng ere, m e grph, o eleng he ere lo elee ll ege nen o. 2. Deleng ere or n ege n nree he nmer of omponen. In f, eleng n ege n only nree he nmer of omponen y 1, eleng ere n nree y mny (oner he lqe K 1,m ) Defnon. A -ege or -ere of grph n ege or ere hoe eleon nree he nmer of omponen. We re G e or G M for he grph of G one y eleng n ege e or e of ege M. We re G or G S for he grph of G one y eleng ere or e of ere S. An ne grph grph one y eleng e of ere. We re G[T] for G T,here T = V(G) T; h he grph of G ne y T. Remrk. 1. When T V(G), he ne grph G[T] on of T n ll ege hoe enpon re onne n T. 2. The fll grph elf n ne grph, re nl ere. 3. A e S of ere n nepenen e f n only f he grph ne y h no ege Emple. The grph elo h -ere n y. I -ege re qr,, y, n yz. r z y q p

4 Grph heory 1.2 Ph, Cyle, n Trl 1 emeer Th grph h C 4 : n Ρ 5 : grph no ne grph. The grph ne y {,,, } ke. Th grph no n ne grph y {,,, }. The grph Ρ 4 oe or n ne grph; he grph ne y {,,, }, lo y {,,, }. Eere Le e ere of onnee mple grph G. Proe h h neghor n eery omponen of G. Conle h no grph h -ere of egree 1. Proof. Sne G onnee, he ere n one omponen of G m he ph n G o eery oher omponen of G, n ph n only lee omponen of G. Th h neghor n eery omponen of G. If G h k omponen, k 2, hen G () k, hh mple here no -ere of egree 1. Eere Coner he p: Mml ph re,, ;,,, ;,,, (o re mmm ph). Mml lqe re {,, {,, } (one mmm lqe). Mml nepenen e re: {}; {, }; {, }(o re mmm nepenen e) Theorem. An ege n grph G -ege f n only f elong o no yle. Proof. Le e e n ege n grph G(h enpon, y), n le H e he omponen onnng e. Sne eleon of e ffe no oher omponen, ffe o proe h H e onnee f n only f e elong o yle. ( ) Sppoe h H e onnee. Th mple h H e onn n, y-ph, n h ph omplee yle h e. ( ) Sppoe h e le n yle C. Chooe, V(H). Sne H onnee, H h, -ph P. If P oe no onn e, hen P e n H e, o H e onnee. If P onn e, ppoe y ymmery h eeen n y on P. C y P e Sne H e onn, -ph long P, n, y-ph long C, n y, -ph long P. The rny of onneon relon mple h H e h, -ph. We h for ll, V(H), o H e onnee Lemm. Eery loe o lk onn n o yle. Proof: We e non on he lengh { of loe o lk W. B ep: { = 1. A loe lk of lengh 1 rere yle of lengh 1. Inon ep: { > 1. Ame he lm for loe o lk horer hn W.

5 Grph heory 1.2 Ph, Cyle, n Trl 1 emeer If W h no repee ere(oher hn fr = l), hen W elf form yle of o lengh. If ere repee n W, hen e e W rng n rek W no 2, -lk. Sne W h o lengh, one of hee o n he oher een. o een The o one horer hn W. By non hypohe, onn n o yle, n h yle pper n orer n W Remrk. A loe een lk nee no onn yle; my mply repe. Neerhele, f n ege e pper ely one n loe lk. Then W oe onn yle hrogh e. e y Le, y e he enpon of e. Deleng e from W lee n, y-lk h o e. By Lemm 1.2.5, h lk onn n, y-ph, n h ph omplee yle h e Defnon. A pron of G pefon of o jon nepenen e n G hoe non V(G). An X, Y-grph pre grph h pron X, Y Theorem. A grph pre f n only f h no o yle. Proof: ( ) Le G e pre grph. Eery lk lerne eeen he 2 e of pron, o eery rern o he orgnl pre e hppen fer n een nmer of ep. Hene G h no o yle. ( ) Le G e grph h no o yle. We proe h G pre y onrng pron of eh nonrl omponen. Le e ere n nonrl omponen H. For eh V(H), le f() e he mnmm lengh of, -ph. Sne H onnee, f() efne for eh V(H). Le X = { V(H) : f() een} n Y = { V(H) : f() o}. An ege, hn X or Y ol ree loe o lk ng hore, -ph, he ege, n he reere of hore, -ph. By Lemm , h lk m onn n o yle, onron. Hene X n Y re nepenen e. Alo X Y = V(H), o H n X, Y-grph Remrk. Theorem mple h heneer grph G no pre, e n proe h emen y preenng n o yle n G. Th mh eer hn emnng ll pole pron o proe h none ork. When e n o proe h G pre, e efne pron n proe h he 2 e re nepenen; h eer hn emnng ll yle Defnon. The non of grph G 1, G 2,,G k, ren G 1 G 2 G k, he grph h ere e k V(G ) n ege e 1 k E(G ). 1 = =

6 Grph heory 1.2 Ph, Cyle, n Trl 1 emeer Emple. K 4 n e epree he non of o 4-yle Theorem. K n n e epree he non of k pre grph f n only f n 2 k. Proof: We e non on k. B ep: k = 1. Sne K 3 h n o yle n K 2 oe no, K n elf pre grph f n only f n 2 1. Inon ep: k > 1. We proe eh mplon ng he non hypohe. ( ) Sppoe h K n = G 1 G 2 G k, here G pre. Ce n = = 8. We pron he ere e V(K n ) no 2 e X, Y h h G k h no ege hn X or hn Y. V(K 5 ) = {,,,, }, X = {, }, Y = {,, } The non of he oher k 1 pre grph m oer he omplee grph ne y X n y Y. Applyng he non hypohe o eh yel X 2 k-1 n Y 2 k-1, o n 2 k k-1 = 2 k. ( ) Sppoe h n 2 k. We pron he ere e V(K n ) no e X, Y, eh of ze mo 2 k-1. By he non hypohe, e n oer he omplee grph ne y eher e h k 1 pre grph. The non of he h h grph on X h he h h grph on Y pre grph. Hene e on k 1 pe grph hoe non on of he omplee grph ne y X n Y. The remnng ege re hoe of he lqe h pron X, Y. Leng h e he k h pre grph omplee he onron. Eere Non-ne proof of Theorem ) Gen n 2 k, enoe he ere of K n n nry k-ple. Le G e he omplee pre grph h pron X, Y, here X he e of ere hoe oe he 0 n poon, n Y he e of ere hoe oe he 1 n poon. Sne eery o ere oe ffer n ome poon, h K n = G 1 G 2 G k. To llre h: n = = 8, enoe he ere of K 5 000, 001, 011, 101, 110. Le G e he omplee pre grph h pron X, Y, = 1, 2, 3 follo: X 1 = {000, 001, 011}, Y 1 = {110, 101}, X 2 = {000, 001, 101}, Y 2 = {011, 110}, X 3 = {000, 110}, Y 3 = {001, 011, 101} Y 110 X X 1 G 1 Y 2 X 3 1 G 2 G Y 3 3 Κ 5 = G 1 G 2 G 3. K 5 G 1 G 2 G 3

7 Grph heory 1.2 Ph, Cyle, n Trl 1 emeer ) Gen h K n non of pre grph G 1,, G k, enoe he ere of K n n nry k-ple. For 1 k, le X, Y e pron of G. Agn ere he k-ple( 1,, k ), here = 0 f X n =1 f Y or X Y. Sne eery 2 ere re jen n he ege jonng hem m e oere n he non, hey le n oppoe pre e n ome G. Therefore he k-ple gne o he ere re n. Sne he k-ple re nry k-ple, here re mo 2 k of hem, o n 2 k Defnon. A grph Elern f h loe rl onnng ll ege. We ll loe rl r hen e o no pefy he fr ere keep he l n yl orer. An Elern r or Elern rl n grph r or rl onnng ll ege. An een grph grph h ere egree ll een. A ere o [een] hen egree o[een]. A mml ph n grph G ph P n G h no onne n longer ph. When grph fne, no ph n een foreer, o mml(non-eenle) ph e. Emple. The grph G elo h n Elern r,,,,, j,, k, m, n, p, r,, p,, n, j, k,. G : k j m n p r The grph H elo h n Elern rl,,,,,, y,,,, z,. y A mml ph n H :,,,,,, z Lemm. If eery ere of fne grph G h egree le 2, hen G onn yle.(the proof e he ehnqe lle eremly.) Proof: Le P e mml ph n G, n le e n enpon of P. Sne P nno e eene, eery neghor of m lrey ere of P. Sne h egree le 2, h neghor n V(P) n ege no n P. z The ege omplee yle h he poron of P from o. Remrk. The proof of Lemm n emple of n mporn ehnqe of proof n grph heory h e ll eremly. When onerng rre of gen ype, hoong n emple h ereme n ome ene my yel efl onl nformon. For emple, ne mml ph P nno e eene, e on he er nformon h eery neghor of n enpon of P elong o V(P).

8 Grph heory 1.2 Ph, Cyle, n Trl 1 emeer In ene, mkng n ereml hoe goe rely o he mporn e. In Lemm , e ol r h ny ph. If eenle, hen e een. If no, hen omehng mporn hppen Theorem. A grph G Elern f n only f h mo one nonrl omponen n ere ll he een egree. Proof: ( ) Sppoe h G h n Elern r C. Eh pge of C hrogh ere e 2 nen ege, n he fr ege pre h he l he fr ere. Hene eery ere h een egree. Alo, o ege n e n he me rl only hen hey le n he me omponen, o here mo one nonrl omponen. ( ) Amng h he onon hol, e on n Elern r ng non on he nmer of ege, m. B ep: m = 0. A loe rl onng of one ere ffe. Inon ep: m > 0. Ame he lm hol for grph h k (< m) ege n h mo one nonrl omponen n ere ll he een egree. Wh een egree, eh ere n he nonrl omponen of G h egree le 2. By Lemm , he nonrl omponen h yle. Le G e he grph one from G y eleng E(C). Sne C h 0 or 2 ege eh ere, eh omponen of G lo n een grph. Sne eh omponen lo onnee n h feer hn m ege, e n pply he non hypohe o onle h eh omponen of G h n Elern r. To omne hee no n Elern r of G, e rere C, hen omponen of G enere for he fr me e eor long n Elern r of h omponen. Th r en he ere here e egn he eor. C When e omplee he rerl of C, e he omplee n Elern r of G Propoon. Eery een grph eompoe no yle. Proof: In he proof of Theorem , e noe h eery een nonrl grph h yle, n h he eleon of yle lee n een grph. Th h prpoon follo y non on he nmer of ege Propoon. If G mple grph n hh eery ere h egree le k, hen G onn ph of lengh le k. If k 2, hen G lo onn yle of lengh le k+1. Proof: Le e n enpon of mml ph P n G. Sne P oe no een, eery neghor of n V(P). Sne h le k neghor n G mple, herefore P h le k ere oher hn n h lengh le k. If k 2, hen he ege from o frhe neghor long P omplee ffenly long yle h he poron of P from o Propoon. Eery grph h nonloop ege h le 2 ere h re no ere. Proof: If n enpon of mml ph P n G, hen he neghor of le on P. Sne P onnee n G, he neghor of elong o ngle omponen of G, n no -ere.

9 Grph heory 1.2 Ph, Cyle, n Trl 1 emeer Eere Sppoe h eery ere of loople fne grph G h egree le 3. Proe h G h yle of een lengh. Proof. Le P e mml ph n G h n enpon of P. Then h le 3 neghor on P. Le, y, z e 3 h neghor of n orer on P. Coner 3, y-ph: he ege y; y z he ege folloe y he, y-ph n P; n he ege z folloe y he z, y-ph n P. Thee ph hre only her enpon, o he non of ny 2 yle. By he pgeonhole prnple, 2 of hee ph he lengh h he me pry (o or een). The non of hee 2 ph n een yle Remrk. Mmm men mmm-ze n mml men no lrger one onn h one Lemm. In n een grph, eery mml rl loe. Proof: Le T e mml rl n n een grph. Eery pge of T hrogh ere e 2 ege, none repee. Th hen rrng ere oher hn nl ere, T h e n o nmer of ege nen o. Sne h een egree, here remn n ege on hh T n onne. Hene T n only en nl ere. In fne grph, T m nee en. We onle h mml rl m e loe Theorem Seon proof ng eremly rely. Proof: Sppoe grph G h mo one nonrl omponen n ere ll he een egree. Le T e rl of mmm lengh; T m lo e mml rl. By Lemm T loe. Sppoe h T om ome ege e of G. Sne G h only one nonrl omponen, G h hore ph from e o he ere e of T. Hene ome ege e no n T nen o ome ere of T. Sne T loe, here rl T h r n en n e he me ege T. We no een T o on longer rl hn T. Th onr he hoe of T, n hene T rere ll ege of G Theorem. For onnee nonrl grph h ely 2k o ere, he mnmm nmer of rl h eompoe m{k, 1}. Proof: A rl onre een egree o eery ere, eep h non-loe rl onre o egree o enpon. Therefore, pron of ege no rl m he ome non-loe rl enng eh o ere. Sne eh rl h only 2 en, e m e le k rl o fy 2k o ere. We lo nee le one rl ne G h n ege, n Theorem mple h one rl ffe hen k = 0. I remn o ho h k rl ffe hen k > 0. Gen h grph G, e pr p he o ere n G (n ny y) n form G y ng for eh pr n ege jonng 2 ere llre elo. P : G : G : The relng grph G onnee n een, o y Theorem , h n Elern r C. A e rere C n G, e r ne rl n G eh me e rere n ege of G E(G). Th yel k rl eompong G.

10 Grph heory 1.2 Ph, Cyle, n Trl 1 emeer Eere Le G e onnee grph h le 3 ere. Proe h G h 2 ere, y h h G {, y} onnee n, y re jen or he ommon neghor. Proof. Le P e longe ph n G h n enpon. Le e neghor of on P. Noe h P h le 3 ere. If G onnee, le y =. Ohere, omponen off from P n G h mo one ere; ll. The ere m e jen o, ne ohere e ol l longer ph. In h e, le y =. P : Eere Le G e onnee mple grph h oe no he P 4 or C 4 n ne grph. Proe h G h ere jen o ll oher ere. Proof. Coner ere of mmm egree n G. If h nonneghor y, le,, e he egnnng of hore ph o y ( my eql y). y Sne () (), ome neghor z of no jen o. If z jen o, hen {z,,, } ne C 4 ; ohere, {z,,, }ne P 4. z Th m e jen o ll oher ere. Eere Proe h he ege of onnee grph h 2k o ere n e prone no k rl f k > 0 y ng non on k. Proof. If k = 1, e n ege eeen he o o ere, on n Elern r, n elee he e ege o ge rl. If k > 1, le P e ph eeen 2 o ere. The grph G = G E(P) h 2k 2 o ere, ne e he hnge egree pry only he en of P. We pply he non hypohe o eh omponen of G h h o ere. Any omponen no hng o ere h n Elern r h onn ere of P; e pl no P o o hng n onl rl. Alogeher, e he e he ere nmer of rl o pron E(G). Homeork 2: , , , , e on Jne 25.