23 201121 51.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 . . . . . . . . . . . . . . . . . . . . . 81.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 () . . . . . . . . . . . . . . . . . . . . . . 131.3 , . . . . . . . . . 191.6 , , . . . . . . . . . . . . 242 332.1 . . . . . . . . . . . . . . . . . . . . . . . . 332.2 , . . . . . . . . . . . . . . . . 362.3 . . . . . . . . . . . . . . . . . . . . . . . . 413 473.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 Eulerian Hamiltonian 614.1 Euleiian . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2 . . . . . . . . . . . . . . . . 674.3 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 705 753.1 . . . . . . . . . . . . . . . . . . . 733.2 Kuiatowski . . . . . . . . . . . . . . . . . . . . . . . . . 733.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.4 . . . . . . . . . . . . . . . . . 7334 11.1 G G = (V, E) V ( ) E . e E ( ). e u v e = {u, v} u v . ( ) , ( ) . V(G) E(G) , . - n m. n = |V(G)| m = |E(G)|. n m, . v V(G) v v . v N(v) N(v) =def {u | {v, u} E(G)}. S V(G) :N(S) =defvSN(v) S. v N[v] =def N(v) {v}. v V(G) N(v) = v , N[v] = V(G) v . e E(G) e = {u, v} u v e e u v ( 36 1. u, v). u = v . . - . . 1.1. 1.1....vI ..v2..v3.. v4..v..v6() G...vI ..v2..v3.. v4..v..v6() H 1.1: G H. () V(G) = {vI, v2, v3, v4, v, v6} E(G) = {{vI, v2}, {v2, v3}, {v3, v4}, {v4, v}, {v, v6}, {v6, vI}, {v2, v6}, {v3, v}} () V(H) = {vI, v2, v3, v4, v, v6} E(H) = E(G) {{v2, v}, {v3, v6}}. - . , . {v2, v} {v3, v6} . 1.1. 1.1() - . . . 1.1. G H 1-1 f : V(G) V(H) {u, v} E(G) {f(u), f(v)} E(H).1.1. 7 G G H. G H G H. ....a .. c..b..d.. e() G...3 ..I..2....4() K3,2 1.2: G K3,2. 1.2. 1.2 . 1-1 f :f(a) = I, f(b) = 2, f(c) = 3, f(d) = 4, f(e) = , G K3,2. G K3,2. . 1.1. . 1.2. G H ( 1.8) G H . 1.1. . - . - . - - . - [19]. Jennei,Kblei, McKenzie Toian [13].8 1. 1.2 . - . ( ) ( ) . - , . . . . . . , G V(G) = {vI, v2, . . . , vn} nn A = [ai,j] ai,j =_I {vivj} E(G)0 {vivj} / E(G). A G. 1.2. n2, - (- ). vi vj - , O(I) ai,j. n! . . G V(G) = {vI, . . . , vn} E(G) = {eI, . . . , em} n m B = [bi,j] - . B :bi,j =_I ej vi0 ej vi. B G.1.2. 9 1.3. - 1.2; - . V(G) - . () . V(G) - - . . v V(G) - . v V(G) - N(v) . N(v) . u N(v) u u . u u . 1.3. 1.1() :A =________0 I 0 0 0 II 0 I 0 0 I0 I 0 I I 00 0 I 0 I 00 0 I I 0 II I 0 0 I 0________B =________I 0 0 0 0 I 0 0I I 0 0 0 0 I 00 I I 0 0 0 0 I0 0 I I 0 0 0 00 0 0 I I 0 0 I0 0 0 0 I I I 0________ 1.3 1.1(). . - Spiniad [21], - .10 1. ..vI.v2.v3.v4.v.v6.. v2.. v6..vI..v3..v6..v2..v4..v..v3..v..v3..v4..v6..vI..v2..v 1.3: 1.1().1.3 G v V(G) v. degG(v) degG(v) = |NG(v)| , deg(v) =|N(v)|. (G) = minvV(G)deg(v) (G) = maxvV(G)deg(v),. G d(G) = InvV(G)deg(v), (G) = mn .1.3. 11 0 1 . r, ( v V(G), degG(v) = r) G r- . 1.4. 1.1 -: (G) = 2, (H) = 2 (G) = 3, (H) = 4, . d(G) = 2.66, d(H) = 3.33 (G) = I.33, (H) = I.66, . 1.1. G n m . :1. vV(G) deg(v) = 2m.2. (G) d(G) (G).3. (G) = d(G)2 .. . - . 1. deg(v) , , v deg(v) . 1) - . 1 (). B 2m. , ( v) 1 v. deg(v) B deg(v). - , . 1.1. .. VI V2 - . V(G) , V(G) = VI V2. 1.1 :vV(G)deg(v) =vVIdeg(v) +vV2deg(v) = 2m.12 1. vV2 deg(v) , 2m vVI deg(v) . deg(v) vVI deg(v) , |VI| . 1.1. r- rn = 2m.. 1.1 r. 1.4. n ; 1.5. n ; -. . 1.2. = dI, . . . , dn dI dn - G n vI, . . . , vn dI, . . . , dn,. G . = dI, . . . , dn G 1-1 : V(G) {I, . . . , n} degG(v) = d(v). - . . . - . . - . = dI, . . . , dn :1. i = I, . . . , n, 0 di n I. - ( 1.4) .2. di 1.1 ( - dI + + dn ).1.3. 13 1.5. 4, 3, 2, 2, I, I , 4, 4, 3, 2 . . ' - G G v dv - dv ( ). 1.6. = 3, 3, 2, 2, 2, 2 = 2, I, I, 2, 2. : = 2, 2, 2, I, I. = 2, 2, 2, I, I = I, I, I, I . 1.2. = dI, . . . , dn, n = d2I, . . . , ddI+II, ddI+2, . . . , dn, nI .. = dI, . . . , dn . , G vI . vI, d2, . . . , ddI+I. - - vi, vj vI (i, j > I) di > dj {vIvi} / E(G) {vIvj} E(G). di > dj vk = vI {vkvi} E(G) {vkvj} / E(G). G G - {vIvj}, {vkvi} {vIvi}, {vkvj}. - G . G vI G di dj > 0. G . - vI d2, . . . , ddI+I. G = G vI vI . , G = d2I, . . . , ddI+II, ddI+2, . . . , dn. G vI vI dI - . G vI dI, dI' v G = (V, E) - v v .14 1. , - . G . . : n : , 1. d : d n 2. d : d < 0 3. di = 0 i = I, . . . , n 4. 3. dI 6. dI 7. n := n I.8. 1. - , . . dI - dI - = kI. - I Gk, GkI, . . . , GI. 1.7. .1. = 4, 3, 3, 2, 2.2. = 4, 4, 3, 2, I.1.4. () 13 - . - ( ). 1.3 ([8]). = dI, . . . , dn, I r n ri=Idi r(r I) +ni=r+Imin{r, di}. Tiipathi Vijay (2003) [22] r r {I, . . . , n}. - 1 ( ) [2]. - [10].1.4 () , . . . - . 1.3. n I , Kn :Kn =_{vI, . . . , vn}, {{vi, vj} | I i, j n}_. 1.4() 1.4(). 1.4. G V(G) = {vI, . . . , vn} {vi, vi+I} E(G) . n 2 Pn = {{vI, . . . , vn}, {{vi, vi+I} | I i < n}} .16 1. .....() K4......() K 1.4: . . vI, vn . (x, y)-o - x y . G V(G) = {vI, . . . , vn} {vi, vi+I}{vI, vn} E(G) . n 3 Cn = {{vI, . . . , vn}, {{vi, vi+I} | I i < n} {vIvn}} . . C3 K3. Cn , n. ( ) ( ). ( ) ( ) . - - vi, vi+I - . , - vi, vi+I vI, vn. 1.8. G 1.1(). - : P4 = {vI, v2, v3, v4}, P4 = {vI, v6, v, v4} : P = {v3, v4, v, v6}, P = {v3, v, v6, v2} C4. 1.3.1.4. () 17.....() P4......() P.....() C4......() C 1.3: . 1.5. G = (V, E) - V(G) - , V(G) = A B A B = e = {x, y} E(G), x A y B. A, B - G , G = (A, B, E). , . G = (A, B, E) x A y B {x, y} (G). G = (A, B, E) Kp,q p = |A| q = |B|. p = I KI,q .........() K3,4.......() G = ({}, {}, E) 1.6: . - 1.6() C6 1.6. . 1.6. 1.2. .18 1. 1.6. X = {xI, . . . , xp} Y = {yI, . . . , yq}. Rp,q _X Y,_{(xi, yj), (xk, yl)} | |i k| +|j l| = I__. - ...................... 1.7: R4,. 1.7 R4,. 1.7. G = (V, E) . G - L(G) :L(G) =_E(G),_{e, e} | e, e E(G) e e = __. L(G) G L(G) - .... eI,2 ..e2,3..e3,4.. e4,..e,6..eI,6.. e2,6.. e3, 1.8: L(G) G 1.1(). 1.8 G 1.1().1.3. , 19 1.6. L(G) -:1. G 1.1().2. G P.3. G C.4. G K.3. G K3,4. 1.7. Rp,q p, q I . 1.8. (G) (G) p, q I :1. G Kp,q.2. G Rp,q.1.9. Kn n.1.5 , . . 1.8. G = (V, E) G = (V(G), {{u, v} | u, v V(G) {u, v} / E(G)}) . G G. 1.3. G, G G. G = (V, E) v V(G). v v v. G v ,G v = {V {v}, E(G) {u, v} | {u, v} E(G)}.20 1. e E(G) G e. - . S V(G) GS S. E E(G) GE E. 1.9. G = (V, E) e = {u, v} E(G). - G/e u, v e w N(w) = N(u)N(v). 1.10. G = (V, E) v V(G) u w. G/v v {u, w}. 1.11. G = (V, E) e = {u, v} E(G). e w N(w) = {u, v}. G e. e P3. P2. H G H G. 1.9 . 1.10. . A, B AB = . . G H V(G) V(H) = . G H E(G) E(H) = . 1.12. G H . G H :G H = (V(G) V(H), E(G) E(H)) . 1.13. G H . G H :G H = (V(G) V(H), E(G) E(H) {{u, v} | u V(G), v V(H)}) . G H - V(G) V(H).1.3. , 21...a.. b..d..c.C4...a.. b..c..d.C4 2K2....a..d..c.C4b. ...a.. b..d..c.C4{b, c} P4. ...a.. b..w.C4/{d, c} K3. ...a..d..c.C4/b K3. ...a.. b.. w..d..c.C4 {a, b} C. 1.9: , /, / . 1.14. G H . G H :G H = (V(G) V(H), {{(u, x), (v, x)} | {u, v} E(G) E(H)}) . G H : G H, v V(G) Hv {u, v} G Hu Hv Hu Hv 1-1 . 1.10 . 1.11 C4P3. 1.4. , .22 1. .........C4 P3..........C4 P3...............C4P3. 1.10: , . . G k 0. ,k G =def G G. .k G(k)=def G G. .k G[k]=def G G. .k . 1.1() 1.1() - . . 1.15. G H . H - G V(H) V(G) E(H) E(G). H G. G H. - . 1.5. G . 1.16. G H . H - G V(H) V(G) u, v V(H), {u, v} E(H) {u, v} E(G).1.3. , 23.........C4.P3. . =............ 1.11: C4P3.S V(G). G[S] :(S, {{u, v} | u, v S {u, v} E(G)}) . G[S] G[S] = G {V(G) S}. 1.6. G . 1.17. G H . H - G V(H) = V(G) E(H) E(G). 1.7. G . 1.8. H G H G. 1.9. : C4P3 C4 C4 C4. Cn n 3 PnI. p, q 2 Kp,q C4. 1.18. F , -. F , - , G F G G G F.24 1. 1.11. Rp,q -; 1.12. Rp,q. k Rp,q Ck . 1.13. G . G . 1.14. Kp,q Kp Kq.1.6 , , G. G () W = [vI, . . . , vr] I i < r,{vivi+I} E(G). vI vr (vI, vr)-. -. vI vr - W (vI, vr)- G r. - . ( ) - W = [vI, . . . , vr, vI] G. - . , . 1.9. . , . 1.10. G 1.1(). W P v3 v6 : W = [v3, v, v4, v3, v2, v6] . P = {v3, v4, v, v6} 3.1.3. G x, y V(G). G (x, y)- (x, y)-.1.6. , , 23. 1.9. - : G (x, y)- W x y W. W = [vI, . . . , vr] vI = x vr = y . W y W. W = [vI, . . . , vrI]. W r, P vI vrI W ( ). vI vrI vr {vrI, vr} P. vI(= x) vr(= y) W . 1.19. x, y G. dist(x, y) x y () - x y. dist(x, y) = . 1.11. G 1.1(). dist(v3, v6) = 2 P = {v3, v, v6} P ={v3, v2, v6}. dist(vI, v4) = 3 P = {vI, v2, v3, v4}. 1.1 ( ). u, v, w - G :dist(u, v) + dist(v, w) dist(u, w). dist(u, v) = dist(v, w) = -. dist(u, v) - (u, v)- Puv. dist(v, w) - (v, w)- Pvw. Puw = PuvPvw - (u, w)-. . 1.4. .26 1. . . G = (A, B, E) A B = e E {a, b} a A b B. G Ck ( ). Ck = {vI, . . . , vk}. vI A. v2 B, v3 A . vi A i vi B i . vk A k . vI, vk A, {vI, vk} E . G. G . G , V(G). v V(G). - v. A v -. dist(v, a) a A. B v . dist(v, b) b B. v 0 v B. A B = . G = (A, B, E) . e {ai, aj} ai, aj A. CA G v :CA = {v, . . . , ai. ., aj, . . . , v. .}, |CA| = +I+ = G. , e {bi, bj} bi, bj B. CB G v :CB = {v, . . . , bi. ., bj, . . . , v. .}, |CB| = +I+ = . G A B G . 1.20. G k I. G Gk= (V(G), {{u, v} | dist(u, v) k}) .1.6. , , 27 GI G........P : ......P2 :......P3 : 1.12: 3 P. 1.12 . 1.15. k Gk Kn () G Pn () G Cn. 1.16. G v G. G v G2; 1.21. v G :ecc(v) = maxuV(G)dist(v, u). :dia(G) = maxvV(G)ecc(v). :iad(G) = minvV(G)ecc(v). - . 1.22. G v V(G). iad(G) = ecc(v) v cent(G) G. dia(G) = ecc(v) v fai(G) G.28 1. ...3 ..2..2.. 3..2..2... ....4.. 3.. 3..4.. ..4.. 1.13: . fai(G) ( ) cent(G) ( ).1.13 cent(G) fai(G). 1.10. Kn . Kp,q p, q 2.1.17. . , . 1.4. G, rad(G) dia(G) 2 rad(G).. (min max). x, y G v G. dist(v, x) ecc(v) dist(v, y) ecc(v). ( 1.1) :dist(x, y) dist(v, x) + dist(v, y). v , ecc(v) = iad(G). ,dist(x, y) dist(v, x) + dist(v, y) 2 ecc(v) = 2 iad(G). - . 1.5. G, cent(G) = far(G) = V(G) cent(G) far(G) = .1.6. , , 29. x cent(G) fai(G). dia(G) =ecc(x) iad(G) = ecc(x). cent(G) fai(G) = iad(G) =dia(G). v V(G) iad(G) ecc(v) dia(G). v V(G), ecc(v) = iad(G) = dia(G) cent(G) =fai(G) = V(G). . 1.23. G - : cim(G). gii(G). 1.24. C = (vI, . . . vk, vI) {vi, vj} j / {i I, i +I}. ., P = (vI, . . . vk) {vi, vj} j / {i I, i + I}. . . -. 1.11. . 1.12. G 1.1() -: cim(G) = 6 gii(G) = 3. C = (v2, v3, v, v6, v2) C4. P = (vI, v2, v3, v4) P4. G[{v2, v3, v, v6}] 4 G. 1.6. G (G) crm(G) I.. P = (vI, . . . , vk) G . vI . u N(vI) P. = {u} P {u, vI} E(G) P . 30 1. P ( ) P. N(vI) P. deg(vI) = |N(vI)| (G), (G) +I N[vI]. (G) + I cim(G). 1.5. G (G) I .. m n . - n. n 3 m 3 . G |V(G)| nI. |V(G)| = n. (G) 2 1.6 . (G) I v deg(v) I. G v n I (G v) I . . - . - [3]. - [4] [12] . 1.18. G H dia(G) = p dia(H) = q. dia(G H); 1.19. Rp,q. 1.20. G dia(G) 2. G G.1.6. , , 31 G = (V, E) n = |V(G)| m = |E(G)| N(v) vdegG(v) vdeg(v) = 0 deg(v) = I deg(v) = n I (G), (G) G(G) = mn Kn, Pn, Cn , Kp,q p + q G[S] G ecc(v) dia(G), iad(G) cent(G) fai(G) dist(u, v) u vcim(G) gii(G) 1.1: 1.32 1. 22.1 dia(G) = , x, y . 2.1. G u, v V(G) (u, v)- G. K2 . G v degG(v) I. v - G. Gv G . . 2.1. . G v G , deg(v) I. G G -. 2.1. G G.. () 1.3. () G - . [vI, . . . , vn] . G Pi (vi, vi+I) i = I, . . . , n I. . Pi 3334 2. Wi. WI, . . . , WnI -. 2.1. G, G G -.. n G. K2 2KI 2KI = K2. nI . n . G n - v V(G). H = Gv H H . v , NG(v) = V(G) {v}, G -. v G G v G . - x, y H {v, x} E(G) {v, y} / E(G) ( v, x v, y -). H {v, x} G - G, H {v, y} G G. 2.2. G (maximal) G . . - - . . - : u v (u, v)- G. - . x, y, z x y y z x z. y z (y, z)- - (x, y)-. ( y) x (x, y)-. - (x, z)-.2.1. 33 G G[S] S . 2.1. H G - G. 2.2. . 2.3. H G (H) (G) (H) (G).: (H) (G) (H) (G). (H) < (G). - v G degG(v) = (G) w V(G) degG(v) degG(w). x H degH(x) = (H) y V(H) degH(x) degH(y) H G degG(x) degG(y). degG(v) > degG(x) - . 2.2. G (G) n2 G .. G H . |V(H)| n2. (H) |V(H)| I < n2. (H) < n2 (G). 2.3 . 2.3. G m n I.. G . - G , m < n I, H n(H) < n m(H) n(H) I. 1.1 (G) (G)2 . (G) 2 (G) I m n . (G) I. G - . v degG(v) = I. H = G v. H 1 - H G. m(H) n(H) I. m(H) = mI n(H) = n I. m n I .36 2. 2.4. k Kk G. 2.5. .2.2 , , - G e . 2.3. e G G. - G e G. 2.4. v G - G. G v G. - . . . - . . 2.1. G V(G) - .. () G . V(G) - A B. a A b B. G (a, b)-. A B .2.2. , 37() G . H G. H G H. A = V(H) B = V(G) A. , . . . 2.4. G G.. e = {x, y} H G. H. H e e H.() He . - (x, y)-. e H e.() H. - u, v V(H). (u, v)-P H e ( H e , He). - H (u, v)- P H . : e (u, v)- P. P He . e (u, v)- P. - H e : (u, x)- Px (y, v)- Py. e H (x, y)- Pxy H e. - PxPxyPy (u, v)- H e . - . - .38 2. 2.5. G - () G. -. . . . 2.2. G k. G k. k 2 G k + I.. (u, v)- P G. P . u P. u k k ( u) P. P k +I k. k 2. u ( P). w ( u) P u. u w u k + I . 2.5. - G . G 1 ( ).. (u, v)- P. v 1. , , - v 1. (- -) P . v 1. v 2. w P v w w P v. w w - P v G . 2.3 -.2.2. , 39 2.1. - G (G) 2. G . .2.6. - G .. (u, v)- P. , u v . P , u P v P. u v G{u, v}. u v . . - . 2.6. G k S V(G). S G S k . S S . S ( -). a, b V(G), S (a, b)- a b G S. v G {v} G. 2.7. k 2 (, ).40 2. ...a ..b.. d..c..e..f..g..h 2.1: G. V(G) {a, h} G. a h G. 2.1. 2.1 G , . {b, c, d} G . {e, f, g}. {b, c} - (a, d)- ({b} {c}) . {e, f}, {f, g}, {d}. {d} - d . - a h ( (a, h)-) PI = [a, e, g, h] P2 = [a, f, h]. (u, v)- u {a, b, c} v {e, f, g, h}. ; - Mengei [17]. Diiac [7]. 2.7 (Mengei). G s, t - (s, t)- G (s, t)- G. Mengei -: ( -) ( (s, t)-). 2.6. - .2.3. 412.3 . 2.8. G n 2 - S G |S| 2. 2.9. G G - G K2. 2.2. G 2.1 - d. G[{d, e, f, g, h}] - G[{a, b, c, d}]. G. - . 2.8. G . G - .. G 1, v, x, y G v. (x, y)- v( v ) x y - . . G x, y V(G). dist(x, y). dist(x, y) = I. e ={x, y}. G e. 1 1, . - G x . 2.1 degGx(y) I G e . G x G e. G e - x y G e e .42 2. ...x .. w.. y.P2.PI...x .. w.. y.P2.PI.R...x .. w.. y..t.P2.PI.R 2.2: 2.8. x, y dist(x, y) < k. x, y G dist(x, y) = k 2. w k x y. {w, y} G. PI P2 G x w. PI, P2, PI, y : PI PI y P2 = P2 {w, y}. G x y. PI, P2 y. (x, y)- R w G G w . : R PI, P2. R PI {w, y} . R PI, P2. t R. t PI. -: PI PI x t - R t y ( PI P22.3. 43 t), P2 = P2 {w, y}. - PI, P2 G, (. 2.2). . 2.2. , . 2.2. . x, y, z V(G), y x z.. G w x z. - 2.2 G . 2.8 G - PI, P2 w y. (PI P2) {w}....xI ..x2..v.. u..v..u.HI.H2 2.3: 2.3. 2.3. HI H2 |V(HI) V(H2)| 2. HI H2 .. {u, v} S = V(HI) V(H2). 2.8, xI, x2 H = HI H2 . xI x2 HI H2. 44 2. xI V(HI) S x2 V(H2) S. 2.2 (u, v)- PI xI. PI PI u, v S ( u = u v = v). 2.2 (u, v)- P2 x2. PIP2 (. 2.3) xI, x2, . 2.4. HI H2 G .. {x, y} S = V(HI) V(H2) w V(HI) V(H2). 2.2 (x, y)- P HI w. H2 P 2.2. H2 H2 P H2 G....x..y.. v ..x..y.Px.Py.P 2.4: 2.4. V(HI) V(H2) = {v}. , , Gv . x, y v HI H2, . P (x, y)- G v. H = HIH2. P Hv E(G) E(H). H = HP - H HI H = HI H2 H. P P HI H2. P HI H2, . x y P x V(HI) y V(H2). P x y v. H 2.8 (x, y)-. , Px(x, v)- HIPy (y, v)- H2, , PxPy P 2.3. 43 (. 2.4). v . -..... ..... ..... ...... ...... ....... 2.3: 2.9. 2.9. - K3 -: , . 2.7. 2.9. 2.8. .2.9. G e E(G). e G Ge . , (), () , 2.1: 2.46 2. 33.1 3.1. . . 1 () . . 3.1. G G .. x, y V(G) PI P2. e = {u, v} PI P2. - H = (PIP2) e H . P H u v. P G e P e G. G G - . G . n G . n = 2 - G. n . n . x, y P G. - P , degG(x) = I. G x . x G G . 3.2. G m = n I.4748 3. . G 2.3 m n I. m n I. m n 1.3 G . 3.1. T n 2 .. x V(T) , degT(x) = I. degV(T)(v) 2 v V(T) \ {x} :vV(T)degT(v) =vV(T)\{x}degT(v) + I 2n + I.vV(T) degT(v) = 2m 2m 2n + I, m n ( m ) 3.2. . - . 3.1. G (G) + I .. (G). (G) 2 3 P3 K2 - G. (G) k I k. (G) = k. G (G) k T k + I - G T. v V(T) T y G. T = T v G = G y |V(T)| = k (G) k I. T G. T G T G. u v T. u V(T) u = (u) G. - v ( ) G. degG(u) k I : degG(u) k. u x V(G) T |V(T) \{u}| =k I. (v) = x. - {u, v} E(T) {(u) = u, (v) = x} E(G) T G.3.2. 49 degG(u) = k I. u y V(G) degG(u) = degG(u) = k I (G) k. (v) = y. {u, v} E(T) {(u) = u, (v) = y} E(G) T G. . 3.2. . .. , , - 2.4 . , , {x, y} . {x, y}- , 3.1 . 3.1. 3 3. 3.2. . 3.3. G nm . 3.4. T (T) k k . 3.5. () = dI, . . . , dn dI + +dn = 2(nI) . n .3.2 3.2. G G. 3.3. G - .30 3. . G . G , T ( - 3.2). V(T) V(G). v V(G) V(T) u V(T). G (v, u)-P G. P v = x0, xI, . . . , xk = u. - v / V(T), {x0, xI} T. {xi, xi+I} i 0 T. xi+I V(T) xi / V(T). xi {xi, xi+I} T T - T ( T ). . G T ( G) - T ( G). G . 3.3. G n = mI .. 3.3 G T. 3.2 T n + I . G (m = n + I) T G T . ( ) - n . . . 3.3. (T, ) T n : V(T) {I, . . . , n} 1-1 V(T) 1 n. v (v) n . - (T, ) (T, ) : T T v V(T): (v) = ((v)). ( ) Pifei Pifei [20].3.2. 31 3.4. Pifei n n 2 1 n.3.1. [, , 2, 3, 3, 2, 8, 8] Pifei I0. . 3.1. n I nn2 Pifei - n. -. 1889 Aithui Cayley [3] . 3.4 (Cayley). nn2.. 1-1 Pifei n n . (). A = [aI, . . . , an2] Pifei n. (T, ) : Pifei : Pifei A = [aI, . . . , an2] n: (T, )1. S = {I, . . . , n}2. T = (V, E) V E = 3. : V S 1-1 S4. |S| > 2 x S AS = S {x}y = aIE = E {I(x), I(y)}A = A {y}3. E S6. (T, )32 3. (T, ). - Pifei A n : Pifei: (T, ) n : Pifei A = [aI, . . . , an2] n1. A = 2. |V(T)| > 2 v T w () v T (w) AT = T v3. A . - A (T, ) - A (T, ). - 3.1. 3.2. 3.1 - Pifei. . Pifei T :{I, }, {4, }, {, 2}, {6, 3}, {7, 3}{3, 2}, {2, 8}, {9, 8}, {I0, 8}. Pifei Pifei ( v 2) :{I, 4, , 6, 7, 3, 2, 9}. 3.2. nn2 n . 3.3. Kn nn2.3.2. 33...I..2..3..4....6.. 7..8..9..I0 3.1: PifeiA = [, , 2, 3, 3, 2, 8, 8]. ( ) , . - - - . - . -, G. T G e G T. - e T - Ce T 3.2. e = e Ce T G. - . - - . 3.5. G D(G) n n :D(G)[i, j] =_ degG(vi), i = j0, . . 3.6. G 34 3. ...vI.. v2..v3..v4.G:...vI.. v2..v3..v4...vI.. v2..v3..v4...vI.. v2..v3..v4...vI.. v2..v3..v4...vI.. v2..v3..v4...vI.. v2..v3..v4...vI.. v2..v3..v4...vI.. v2..v3..v4 3.2: G G. L(G) :L(G) = D(G) A(G), A(G) D(G) - G. n n B i I i n B i i B. Bi|i. . Kiichho[14] [16] . 3.5 (-). G n . i, I i n, G det (L(G)i|i). 3.3. G 3.2 - :L(G) =____2 I I 0I 3 I II I 3 I0 I I 2____3.3. 33 G - 3.3 .det (L(G)4|4) =2 I II 3 II I 3=2 I II 3 I3 0 4=2 3/4 I II 3/4 3 I0 0 4= 4/4 I7/4 3 = 4 2 = 8 -1 . 3/4 . 3.6. Pifei = [2, I, I, 3, , 7]. 3.7. Pifei KI,p. 3.8. G G Kp,q.3.3 -. - . (-) G . (x, y) E(G) x y. :36 3. 3.7. - v V(G) - v degG(v)., o - v V(G) - v deg+G(v).,degG(v) = |{(u, v) E(G)|u V(G)}| deg+G(v) = |{(v, u) E(G)|u V(G)}| . - . 3.8. r, , v = r (r, v)- . . 3.6. - 0 - 1.. T r . - r - . w r ( degG(r) > 0) (r, w)- . w degG(w) = I. (r, w)- degG(w) I. - u, v (u, w) E(T) (v, w) E(T). (r, u)- (r, v)- (u, w) (v, w) . . T - 0 - 1. T , r T. w. (r, w)-. degG(w) = I - uI (uI, w) E(T). u2 (u2, uI) E(T). T 3.3. 37..a.b.c.d.e.f.g.h.i.j.k.l.m.n..a.b.c.d.e.f.g.h. i.j.k.l.m.n..a.b.c.d.e.f.g.h.i.j.k.l.m.n 3.3: . . r (r, w)-, . , . 3.9. ( ) u T u r T. (r, u)-. 3.10. T T. (u, v) E(T) u v v u. w v v () (r, w)-. , v w.38 3. 3.11. T m- m 2 . T m- m . 3.4 . ............ .............. 3.4: . 3.7. m- mk k.. k. k = I m. k I mkI. k. k I. k I mkI. m- mkI m. k m mkI= mk, . 3.9. m- T h n . h + I n mh+IImI . 3.10. 2- . () h 2h+II .3.3. 39, (T, ) Pifei - deg(v), - deg+(v), , , () m- 3.1: 3.60 3. 4Eulerian Hamiltonian4.1 Eulerian G ( ) W = [vI, . . . , vr] I i < r, {vivi+I} E(G). G ( ) . 4.1. G G Euler. G Eulerian. Euleiian... () Euleiian. 4.1. 1.1() Eulei 4.1. 1.1() Eulei 4.1. H - 4.1 Euleiian . Knigsbeig 18 Eulei [23]. - Leonhaid Eulei [1] . Euleiian . .6162 4. EULERIAN HAMILTONIAN ...vI ..v2..v3.. v4..v..v6.(1).. (8)...(6).. (4).(2)...(3)..(10).(9)..(3)..(7). 4.1: H 1.1() Eulei. - Eulei . - W = [v3, v2, vI, v6, v2, v, v6, v3, v, v4, v3] Eulei . 4.1 (Eulei 1736, [9]). G Euler .. . G Eulei W = [vI, . . . , vn, vI] . v W k v W. W v - v. W v k - 2k . G W, deg(v) = 2k, v ....v..x..y.... ...v..w..x..y.... 4.2: v - 4.1. :p(G) =vV(G)(deg(v) 2) = 2(mn). p(G) = 0 G Cn m = n, G v V(G), deg(v) = 2. 4.1. EULERIAN 63G p(G) . p(G) > 0 v 4. G : v . - {v, x} {v, y} v - w x y. v . {v, x} {v, y} x, y - G v ( G) w x y. G (. 4.2) - G w degG (w) = 2 degG (v) = degG(v) 2. p(G) < p(G) n(G) = n + I. - (p(G) ) Eulei W G. W G - w v Eulei W G. G Euleiian. Eulei.1. Euleiian.2. 2 G Cn Cn Eulei.3. deg(v) 4 w deg(w) = 2 4.1.4. G Cn - 2. 4.2. 4.3 Eulei G 1.1() 4.1. G CI0 Eulei G W = [v2, vI, v6, v, v4, w3, w6, w2, w, v3, v2]. wi vi Eulei G:W = [v2, vI, v6, v, v4, v3, v6, v2, v, v3, v2].64 4. EULERIAN HAMILTONIAN ...vI ..v2..w2..v3.. v4..v..v6.G...vI ..v2..w2..v3.. v4..v..v6..w6.G...vI ..v2..w2..v3.. v4..v..w..v6..w6.G...vI ..v2..w2..v3..w3.. v4..v..w..v6..w6.G 4.3: 4.1 1.1(). G v3 - {w2, w, w6} G v3. G v3 w3 . 4.2. G G Euler. G semi-Eulerian . Euleiian , semi-Euleiian. 4.3. 4.4 Eulei. - 1.1() (Euleiian ) . 4.1. 4.1. semi-Eulerian - .. () G semi-Euleiian - Eulei P = [vI, . . . , vk] G.4.1. EULERIAN 63...vI ..v2..v3.. v4..v..v6.(1).. (8)...(6).. (4).(2)...(3).(9)..(3)..(7). 4.4: semi-Euleiian . - Eulei. Eulei : P = [v3, v2, vI, v6, v2, v, v6, v3, v, v4]. {vI, vk} W = [vI, . . . , vk, vI] Eulei G+{vI, vk}. G+{vI, vk} Euleiian 4.1 G +{vI, vk} - . {vI, vk} G + {vI, vk} vI, vk , .() G u, v. G u v G + {u, v}. G + {u, v} 4.1 G+{u, v} Euleiian. Eulei W = [vI, . . . , vk, vI] G+{u, v}. vi = u vj = v u, v W. {u, v} = {vi, vj} G+{u, v}, W . vi, vj W(|i j| = I) vi = vI vj = vk ( vi, vj W). {vi, vj} W P vi, vj W vi, vj W :W = [vI, . . . , vi. .=, vj, . . . , vk. .=, vI] P = [vi, . . . , vI, vk, . . . , vj]. P G Eulei. , Eulei P. 66 4. EULERIAN HAMILTONIAN EuleiW . W - EuleiP. , 4.4 {v4, v3} Eulei W 4.3. W = [v2, vI, v6, v, v4. .=, v3, v6, v2, v, v3. .=, v2] {v4, v3} EuleiP = [v4, v, v6, vI, v2, v3, v, v2, v6, v3]. 4.1. Euleiian semi-Euleiian. . 4.2. Euleiian semi-Euleiian. Kp,q Rp,q m- h. 4.3. 4.3 Euleiian semi-Euleiian Eulei Eulei....vI ..v2..v3.. v4..v..v6 4.3: 4.3.4.2. 674.2 () - (). . - . ,, . : - ( ) . - , - 1 k . G Euleiian Eulei () k = I . Euleiian. - . - -. . G = (V, E) 1-1 w : E(G) N, - E(G) . w(H) H G - H. : W G w(W) .68 4. EULERIAN HAMILTONIAN Kwan [13]. . . G Euleiian Eulei W 4.1. W . G Euleiian . 1.1, - . - . . (u, v) {u, v} (u, v)-'. Puv - (u, v)- G =G {u, v} w({u, v}) = w(Puv) = |Puv|. G - . 4.1 G Euleiian. Eulei W G - 4.1. Eulei W W - {u, v} Puv. -: : G = (V, E) w : E(G) N: W E(G) W min(w(W))1. S 2. S = W = Eulei G3. SI, S2: |SI| = |S2| = |S|/2 SI S2 = S' Dijkstia . Dijkstia . O(n2).4.2. 694. u SI v S2 {u, v} / E(G):Puv = (u, v)- G = G {u, v} w({u, v}) = w(Puv)3. W = Eulei G6. u SI v S2:W = W{u, v} Puv 4.4. G 4.6. G {v2, v3, v, v6}. (v2, v) (v3, v6). ...vI ..v2..v3.. v4..v..v6.G.8.. 7...4.. 2.3...3..2.3....vI ..v2..v3.. v4..v..v6.G.8.. 7...4.. 2.3...3..2.3..6(v6)..9(v4, v). 4.6: 4.4. - . {v2, v} {v3, v6} G G . Eulei W 4.3 :W = [v2, vI, v6, v, v4, v3, v6, v2, v, v3, v2].70 4. EULERIAN HAMILTONIAN , :W = [v2, vI, v6, v, v4, v3, v4, v, v6, v2, v6, v, v3, v2]. w(W) ( ) {v2, v6}, {v6, v} () {v3, v4}, {v4, v}, {v, v6}, . 4.4. 4.7....vI ..v2..v3.. v4..v..v6.G.8...1..10.3...1..3.3...4..4 4.7: 4.4.4.3 Hamiltonian Euleiian (-) . . Euleiian - Hamiltonian . ( ) . 4.3. Hamilton G G G. Hamilton G G G. Hamilton Hamiltonian. Hamilton Sii WilliamRowan Hamilton (1837) - [11]. Hamilton 4.8.4.3. HAMILTONIAN 71..................... 4.8: .4.5. 1.1() 1.1() - Hamilton [vI, v2, v3, v4, v, v6]. 4.9 Hamilton G. ..........G.......H 4.9: G H. G - Hamilton. H 4.9 Hamiltonian. Kn Hamiltonian, . 4.1 Euleiian , Hamiltonian . - Hamilton. . . 4.1. k 3, I = {I, . . . , k 2} II, I2 I |II| +|I2| k. ,72 4. EULERIAN HAMILTONIAN i II i + I I2 j I2 j + I II. 4.2 (Oie [18]). G 3 n. G Hamiltonian.. G . x y deg(x)+deg(y) n2 < n, . . Hamilton G. - P = [vI, . . . , vr] r n. - N(vI) P N(vr) P . G C P. {vI, vr} E(G) . {vI, vr} / E(G). N(vI), N(vr) {2, . . . , r I}, |N(vI) N(vr)| r2 n2. |N(vI)|+|N(vr)| n. - 4.1 II = N(vI), I2 = N(vr), k = r i {2, . . . , rI} vi N(vr) vi+I N(vI). - C = [vI, vi+I, vi+2, . . . , vr, vi, viI, . . . , v2, vI] (. 4.10)....vI..v2. ..viI..vi..vi+I..vi+2. ..vrI..vr 4.10: P 4.2. G Hamilton r < n. G . - w G C v C. P w C r +I. P r +I vp, vq v C. P = C \ {v} vp, vq r I. P = [w, v, P] |P| = r + I. P r.4.3. HAMILTONIAN 73 . 4.3 (Diiac[6]). G - n/2 G Hamiltonian.. (G) n2 G Hamiltonian. x, y deg(x) + deg(y) 2(G) n 4.2 . Eueleiian , Hamiltonian . - (NP-) . - [24]. 4.5. Kp.q Hamiltonian. 4.6. 4.1. 4.7. Hamiltonian G H. G H Hamiltonian. 4.8. Hamilton . 4.9. Hamilton Rp.q. 4.10. 4.1, Hamilton Hamilton. - . 4.11. G . G Hamilton . -, Hamiltonian. , .74 4. EULERIAN HAMILTONIAN Euleiian Euleisemi-Euleiian Eulei Eulei Hamiltonian Hamilton 4.1: 4. 5 5.1 5.2 Kuratowski5.3 5.4 7376 3. [1] E. Avcuivi, e works of Leonhard Euler online,http://www.math.daitmouth.edu/ eulei/.[2] M. Biuz:u :u G. Cu:v1v:u, No graph is perfect., Amei. Math. Monthly,74 (1967), pp. 962963.[3] A. C:viiv., A theorem on trees, Quait. J. Math., 23 (1889), pp. 376378.[4] F. R. K. CuUc., Diameters of graphs: old problems and newresults, CongiessusNumeiantium, 60 (1987), pp. 293317.[3] R. Diis1ii, Graph eory, SpiingeiVeilag, 2003.[6] G. A. Div:c., Some theorems on abstract graphs, Pioceedings of the LondonMathematical Society, 2 (1932), pp. 6981.[7] , Short proof of mengers graph theorem, Mathematika, 13 (1966),pp. 4244.[8] P. Evuos :u T. G:ii:i, Graphs with prescribed degrees of vertices, Mat.Lapok., 11 (1960), pp. 264274.[9] L. EUiiv, Solutio problematis ad geometriam situs pertinentis, Comment.Academiae Sci. I. Petiopolitanae, 8 (1736), pp. 128140.[10] S. H:ximi, On the realizability of a set of integers as degrees of the vertices of agraph, SIAM J. Appl. Math., 10 (1962), pp. 496306.[11] W. R. H:mii1o, Account of the icosiancalculus, Pioceedings of the Royal IiishAcademy, 6 (1838), pp. 69.[12] A. I1:i :u M. Rouiu., Finding a minimum circuit in a graph, SIAM J.Comput., 7 (1978), pp. 413423.7778 [13] B. Jiiv, J. Koniiv, P. McKizii, :u J. TovI, Completeness results forgraph isomorphism, J. Comput. Syst. Sci., 66 (2003), pp. 349366.[14] G. Kivcuuoii., Ueber die fraunhoferschen linien, Monatsbeiichte, Akademiedei Wissenschafen, Beilin, (1860), pp. 662663.[13] M.-K. Kw:, Graphic programming using odd or even points, Chinese Math., 1(1962), pp. 273277.[16] S. Locxi, Graphs, Matrices, Isomorphism,http://math.fau.edu/locke/Giaphmat.htm.[17] K. Miciv., Zur allgemeinen kurventheorie, Fund. Math., 10 (1927),pp. 96113.[18] O. Ovi., A note on hamiltonian circuits, Ameiican Mathematical Monthly, 67(1960), pp. 3337.[19] C. H. P:v:uimi1vioU, Computational complexity, Addison-Wesley, 1994.[20] H. Pi0iiiv., Neuer beweis eines satzes ber permutationen, Aich. Math. Phys.,96 (1918), pp. 742744.[21] J. P. Sviv:u, Ecient Graph Representations, AmeiicanMathematical Society,Fields Institute Monogiaph Seiies 19, 2003.[22] A. Tviv:1ui :u S. Vi,:v, A note on a theorem of erdos & gallai., Disci. Math.,263 (2003), pp. 417420.[23] Wixiviui:, Seven Bridges of Knigsberg, http://en.wikipedia.oig/wiki/SevenBiidges of Konigsbeig.[24] , Travelling salesman problem, http://en.wikipedia.oig/wiki/Tiavellingsalesman pioblem.