k

k

ΚΕΦΑΛΑΙΟ 1 G = (V, E) V E V V V G E G e = {v, u} E v u e v u G G V (G) E(G) n(g) = V (G) m(g) = E(G) G S V (G) S G N G (S) = {u V (G)\S v S : {v, u} E(G)} G v S v V (G) N G (v) = N G ({v}) x V (G) N G (x) = x v 1 v 5 v 1 v 5 v 3 v 4 v 3 v 4 v 2 v 6 G v 2 v 6 G G G V (G) = V (G ) = {v 1, v 2, v 3, v 4, v 5, v 6 } E(G) = {{v 1, v 2 }, {v 1, v 3 }, {v 2, v 3 }, {v 3, v 4 } {v 1, v 5 }, {v 2, v 6 }, {v 4, v 5 }, {v 4, v 6 }, {v 5, v 6 }} E(G ) = E(G) {{v 1, v 4 }, {v 2, v 5 }}

G = (V, E) V = {v 1, v 2, v 3, v 4, v 5, v 6 } E = {{v 1, v 2 }, {v 1, v 3 }, {v 2, v 3 }, {v 3, v 4 } {v 1, v 5 }, {v 2, v 6 }, {v 4, v 5 }, {v 4, v 6 }, {v 5, v 6 }} G G G a b c d b e 4 2 5 d e f G a f c G 1 6 3 H G = ({a, b, c, d, e, f}, {{a, d}, {a, e}, {a, f}, {b, d}, {b, e}, {b, f}, {c, d}, {c, e}, {c, f}}) G H = ({1, 2, 3, 4, 5, 6}, {{1, 4}, {1, 5}, {1, 6}, {2, 4}, {2, 5}, {2, 6}, {3, 4}, {3, 4}, {3, 6}}) G V (G) = {v 1,..., v n } n n A = [a i,j ] (i,j) [n] 2 a i,j = { 1 {v i, v j } E(G) 0 {v i, v j } E(G) G 0 n n! 0 1 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 0 A = 0 0 1 0 1 1 1 0 0 1 0 1 0 1 0 1 1 0 A = 0 1 1 1 1 0 1 0 1 0 1 1 1 1 0 1 0 0 1 0 1 0 1 1 1 1 0 1 0 1 0 1 0 1 1 0

G G G H σ : V (G) V (H) x, y V (G) x y {x, y} E(G) {σ(x), σ(y)} E(H) G H G H G H G H a c υ ω 4 b 5 2 χ τ e d G f ϕ G ψ 6 3 G 1 Q 3 Q 3 r 0 K r = ({v 1,..., v r }, {{v i, v j } 1 i < j r}) r r G r G K r

K 6 K 4,3 K 6 K 4,3 p, q 0 K p,q = (A B, E) A = {v 1,..., v p }, B = {u 1,..., u q } E = {{v i, u j } 1 i p 1 j q} K 1,r r 0 r K 3,3 P 3 C 7 P 3 C 7 r 1 P r = ({v 1,..., v r+1 }, {{v 1, v 2 },..., {v r, v r+1 }}) v 1 v r+1 x y (x, y) r 3 C r = ({v 1,..., v r }, {{v 1, v 2 },..., {v r 1, v r }, {v r, v 1 }}) C 3 (6, 4)

V r = {1,..., r} (p, q) (V p V q, {{(x 1, y 1 ), (x 2, y 2 )} x 1 x 2 + y 1 y 2 = 1}). r 0 V r r r Q r = (V r, {{x, y} x, y V r x y }) Q 0 Q 1 Q 2 Q 3 Q i i = 0, 1, 2, 3 G G G V (G) G (G) G 1, 2, 3, 4 1, 4, 3, 2 3, 2, 1, 4 3, 2, 1, 2 (C 4 ) = { 1, 2, 3, 4 1, 4, 3, 2 3, 2, 1, 4 2, 3, 4, 1 3, 4, 1, 2 4, 3, 1, 2 4, 1, 2, 3 2, 1, 4, 3 } (H) = { 1, 2, 3, 4 4, 2, 3, 1 } V (K n ) (K n ) S n n! = (K n ) = S n G C 4 H G C 4 H

H H G (G) S n G (G) G n! G x, y V (G) x y σ(x) = y σ (G) x y G G {1, 3} {2, 4} C 4 {1, 2, 3, 4} G {2} {3} {1, 4} H {1, 4} {2, 3, 5, 6} {7} G x y σ (G) σ(x) = y C r r 3 K r r 1 K r,r r 1 G G V

5 1 2 1,6 2,1 1 1 G 1 G 2 G 3 G 4 G 5 G 6 G 1 G 2 G 3 G 4 G 5 G 6 {,,,,} G 1 E(G 1 ) = {{,}, {,}, {,}, {,}, {,}, {,}, {,}} G 2 E(G 2 ) = {(,), (,), (,), (,), (,), (,), (,)} G 3 E(G 1 ) 5 1 2,1 2 1 1,6 1 G 4 E(G 1 ) = E(G 1 ) {{}, {}, {}} G 5 E(G 1 ) {,} {,} {,} G 6 E(G 1 ) {{,,}, {,,,}} G = (N, E) E = {{x, y} ( N 2) y 2 = x 3 } G = (R, E) E = {{x, y} ( R 2) y 2 + x 2 = 1} 3 G Q 3

A = [a i,j ] 1 i,j r a i,j = (i + j) ( 2) K r/2, r/2 G 1, G 2, G 3 A = [a i,j ] 1 i,j 8 a i,j = (i + j) 2 σ : V (G) V (H) G H S V (G) σ(n G (S)) = N H (σ(s)) S V (G) σ(s) = {σ(v) v S} G m(g) = ( ) n(g) 2 x, y 1 (x, y) P x 1 P y 1 (p, q) 2 p q p q a, b, r C a Q b (r, r) Q r r Q r r 0 G (G) G (G) = 1 G n(g) A

n n

ΚΕΦΑΛΑΙΟ 2 G G G = (V (G), {{x, y} x, y V (G)}\E) G G G G G L(G) = (E(G), {{e 1, e 2 } e 1, e 2 E(G) e 1 e 2 }). a e b a d c e f b G d c f L(G) K 4 L(K 4 ) G H G H G H = (V (G) V (H), {{(x 1, y 1 ), (x 2, y 2 )} ({x 1, x 2 } E(G) y 1 = y 2 ) ({y 1, y 2 } E(H) x 1 = x 2 )}). G H G H = (V (G) V (H), E(G) E(H)) G H = (V (G) V (H), E(G) E(H)).

V (G) V (H) = G H G H G H G H G + H G H G H = {V (G) V (H), E(G) E(H) {{x, y} x V (G) y V (H)}}. G H G H G H G + H G H G H G H K 3 K 1,3 K 3 K 1,3 K 3 K 1,3 K 3, K 1,3, K 3 K 1,3 K 3 K 1,3,, +, k 0 G k G = G } + {{ + G }, G [k] = G } {{ G } k k G (k) = G} {{ G}. k G 0 G G (0) K 0 G [0] K 1 K 1 K 2 G 1 G 2 G 1 K 1,K 2 G 2 G 1 G 2 K 1 K 2 G 1 G 2 G 1 G 2 K 1 K 2 K 1 K 2 G K1,K 2 H G H

G e E(G) v v e v e v V (G) E G (v) V (G) v E E(G) V (E) = e E e E G S V (G) v V (G) E E(G) e = {x, y} E(G) G\S = (V (G)\S, {{x, y} E(G) {x, y} S = }) G\v = G\{v} G\E = (V (G), E(G)\E) G\e = G\{e} G\{x, y} G\e {x, y} e G\{x, y} {x, y} x y G\e e G v V (G) {x, v} {v, y} x y G/v = (V (G)\{v}, E(G)\{{x, v}, {v, y}} {{x, y}}) v G v e e G H G H G e = {x, y} E(G) v V (G) G/e = (V, E ) V = V (G)\{x, y} {v } E = E(G)\E G (x)\e G (y) {{v, u} u N G ({x, y})\{x, y}} e = {x, y} G x y v {x, y} G

u G v e G e G u v G 1 G 2 G 3 f G f G w w G 4 G 5 G 6 T = {\v, /v, \e, /e} T = {\v, /v, \e, /e} \v \e /v /e T = {\v, /v, \e, /e} A T A = G H H A G H G A A T A A = {\e, \v} H A G υπ G H G H G A = {\v} H A G ϵν G H G

A = {\e} H A G πα G H G A = {\e, \v, /v} H A G τπ G H G A = {\e, \v, /e} H A G ϵλ G H G C 4 C 4 C 5 C 5 G S V (G) G[S] = G\(V (G)\S) G[S] = (S, {{v, u} {v, u} S {x, y} E(G)}). G[S] ϵν G G[S] G S E E(G) G[E] = (V (E), E) G[E] G G H H ϵν G H ϵν G H G H πα H G H ϵν H G H τπ G H ϵλ G υπ ϵν πα τπ ϵλ T = {\v, /v, \e, /e} T G { υπ, ϵν, πα, τπ, ϵλ } G G G G G G G

G G G n(g) = 0 1 ( 4) n 3 C n L(C n ) G m(l(g)) ( ) m(g) 2 K p,q + K r,s (K p + K q ) (K r + K s ) K p,q K p + K q K (m) r K m r P p P q (p, q) L(K 4 ) (2 K 1 ) (3) Q r K [r] 2 G (G) = (G) G = {G G L(G)}

G k 1 k G, G [k] G (k) G L(G) G k,r V (G k,r ) r k {v, u} E(G k,r ) v u G k,r K [r] k G L(G) G = C i1 + + C ir i j 3 1 j r G 1 G 5 {\e, /v} G e v G\v G/v G\e G/e G K 1 G n T n k k 1 G k q 2 Q q 2 q G 1 K 5 k K 2,4 (k k) T W r = K 1 C r (n n) r 3(n 2) + (n 3) 2 n 3 3 K 1,3 + K 1,3 πα Q 3 Q 3

K 1,4 ϵλ Q 3 K 1,4 τπ Q 3 K 3,3 K 5 K 5 r (r, r) Q 3 G K 1,r υπ G K 1,r τπ G r (r, r) L(K 4 ) (r, r) K 1,1+ r 2 (r 1) K 2,1+ r 3 (r 1) K 3,r r 3 G 1 = {C r r 3} G 2 = {P i i 0} G 3 = {Q r r 0} ϵλ τπ υπ πα ϵν k A = {δ (G) G P [k] n n 1} B = {δ (G) G P [k] n n 1}

ΚΕΦΑΛΑΙΟ 3 v G G (v) = N G (v). G δ(g) = { G (v) v V (G)} (G) = { G (v) v V (G)} d(g) = 1 n(g) v V (G) G (v) G ϵ(g) = m(g) n(g) G r r (G) = {k K 1,k υπ G}. G v V (G) G (v) = 2 m(g) δ(g) d(g) (G) ϵ(g) = d(g) 2 v G (v)

V 1 V 2 V (G) 2 m(g) = G (v) = G (v) + G (v), v V 1 v V 2 v V (G) v V 1 G (v) V 1 G (G) G v V (G) z(g) = ( (G) G (v)). n(g) z(g) G (G) r = (G) G r G 1 = G G < r G ϵν G 1 z(g 1 ) < z(g) G m G υπ G r z(g) = 0 G m r G ϵ(g) δ(g) 2 δ, ϵ 0 δ ϵ G n = K δ+1 + K n δ 1 δ(g) δ ϵ(g n ) = ( δ+1 2 )+( n δ 1 2 ) n n ϵ(g n ) = n ϵ(g n ) ϵ δ (G) = {k G H δ(h) k }. G H υπ G δ (H) δ (G) G δ (G) n 1 δ (G)

G δ (G) = 3 δ (G) n δ (G) G H δ(h) n δ (G) H n 1 (n δ (G)) = δ (G) 1 G δ(h) n δ (G) n(h) = n(h) n δ (G) + 1 H G δ(h ) δ (G) n(h ) δ (G) + 1 n(h) + n(h ) > n H H v v H G (v) H (v) δ(h ) δ (G) v H G (v) δ (G) 1 G H δ(h) ϵ(g) δ (G) ϵ(g) G n(g) = 1 < n G n(g) = n δ(g) δ(g) δ (G) v G G (v) δ (G) G = G\v E(G ) m(g) δ (G) V (G ) = n(g) 1 δ (G ) ϵ(g ) = E(G ) V (G ) m(g) δ (G). n(g) 1 G υπ G δ (G) m(g) δ (G). n(g) 1 G δ (G) {ϵ(g), δ(g)} G n δ (G) k (v 1,..., v n ) G i=1,...,n δ Gi (v i ) k G i = G[{v 1,..., v i }] (v 1,..., v n ) G i=1,...,n δ Gi (v i ) k H υπ G δ(h) > k v i H (v 1,..., v n ) H υπ G i Gi (v i ) δ(g i ) δ(h) > k (v 1,..., v n ) G v i δ Gi (v i ) > k

G (v 1,..., v n ) v i v i (v 1,..., v n ) > k G i v j, j < i k G i v i v i+1 δ(g i ) > k δ (G) > k α = [d 1,..., d n ] G σ : V (G) {1,..., n} G (v) = d σ(v) α 3 4 3 3 G 1 5 1 3 1 5 3 G [5, 5, 4, 3, 3, 3, 3, 3, 1, 1, 1] α = [d 1,..., d n ] n 2 d 1 1 α = [d 2 1, d 3 1,..., d d1 +1 1, d d1 +2,..., d n ] α = [d 1,..., d n ] G V (G) = {v 1,..., v n } δ G (v i ) = d i, 1 i n f(g) = v N G (v 1 ) v 1 (d 2,..., d d1 +1) v i, v j N G (v 1 ) d i > d j {v 1 v i } E(G) {v 1, v j } E(G) d i > d j v h v 1 {v h, v i } E(G) {v h, v j } E(G) G G {v 1, v j } {v h, v i } {v 1, v i } {v h, v j } f(g ) > f(g) v 1 (d 2,..., d d1 +1) G\v 1 α = [d 2 1, d 3 1,..., d d1 +1 1, d d1 +2,..., d n ] α

G α = [d 2 1, d 3 1,..., d d1 +1 1, d d1 +2,..., d n ] S G d 1 G S G α = [d 1,..., d n ] (d 1,..., d n ) d i r(r 1) + (r, d i ) i=1,...,r i=r+1,...,n ϵ(g) = δ(g) 2 G L(G) n r, s r + s = n s = 0 ( 2) G r s G 2 K 3 ϵλ G G G n m δ(g) m 1 2 (n2 3n + 2).

q, r 1 δ (K 1,q K 1,r ) δ (G) 1 2 ( 2 n(g) 1 ) (2 n(g) 1) 2 8 m(g). G H δ (G), δ (H) k δ (G H) 2k + 1 G δ (G) 1 2 (n 1 (G)) k A = {δ G G P [k] n n 1}, B = {δ G G P [k] n n 1}. α = (d 1,..., d n ) (n d 1 1, n d 2 1,..., n d n 1) α = (d 1,..., d n ) G k G G [k] G (k) k 0

ΚΕΦΑΛΑΙΟ 4 G G W = [v 1,..., v r ] i,1 i<r {v i, v i+1 } E(G) W (v 1, v i+1 ) G r v 1 v r W G[W ] = ({v 1,..., v r }, {{v 1, v 2 },..., {v r 1, v r }}). W = [v 1,..., v r, v 1 ] G n n G (x, y) (x, y) G (x, y) W (x, y) W W = [v 1,..., v r ] G v 1 = x v r = y y W W y i W = [v 1,..., v i ] W (x, y) W W = [v 1,..., v r 1 ] W r G (v 1, v r 1 ) P W v r P {v r 1, v r } (x, y) W

G V (G) = {1,..., n} A = [a i,j ] (i,j) [n] 2 G i i i i = 1,..., n r = 1,..., n a r i,j Ar = [a r i,j ] (i,j) [n] 2 r i j G r r = 1 v i, v j A 1 = A A r 1 = [a r 1 i,j ] ar i,j r 1 v i v j A r = A r 1 A a r i,j = h=1,...,n a r 1 i,h a h,j r v i v j v i v h r 1 v j v h A = C 5 A 2 = 2 0 1 1 0 0 2 0 1 1 1 0 2 0 1 1 1 0 2 0 0 1 1 0 2 A3 = 0 3 1 1 3 3 0 3 1 1 1 3 0 3 1 1 1 3 0 3 3 1 1 3 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 A 4 = C 5 6 1 4 4 1 1 6 1 4 4 4 1 6 1 4 4 4 1 6 1 1 4 4 1 6 x, y G G (x, y) x y G (x, y) G G (x, y) = G G (x) = (G) = G (x, y). y V (G) G (x). x V (G)

β χ Θ (β, χ) (β, χ) (β, χ) Θ x, y G (x, y) = (G) (G) = G (x). x V (G) x V (G) (G) = G (x) x G G (G) x V (G) (G) = G (x) x G G (G) G n(g) 2 G x y K k, k 1 K p,q, p, q 2 p q Q 3 G V (G) G x,y V (G) G (x, y) 0 G (x, y) = 0 x = y x,y V (G) G (x, y) = G (y, x) x,y,z V (G) G (x, y) + G (y, z) G (x, z) G (G) (G) 2 (G)

G H G x, y G v G G (x, v) G (v) G (v, y) G (v) (G) = G (x, y) G (x, v) + G (v, y) 2 G (v) = 2 (G). (C r ) = r 2 = (C r) r 3 2 (P 2 r ) = 2r = (P 2r ) r 1 (C r ) = (C r ) = V (C r ) r 3 (P r ) 2 K 1 r 2 P 2r+1 [(P 2r+1 )] K 2 r 0 (P 2r ) = 1 r 1 G (G) = (G) = V (G) G x (G) = G (x) = (G) v V (G) (G) G (v) (G) G (v) = (G) = (G) (G) = (G) = V (G) G (G) d v V (G) q q (d 1) l 1 l 1 v i = 1,..., l Pv i l v P v τ(p ) Pv 1 = q i, 1 i l 1 Pv i+1 Pv i Pv i G (u) 1 Pv i+1 Pv i+1 P Pv i ( G (τ(p )) 1) Pi v (d 1) i, 1 i r 1 Pv i = Pv 1 (d 1) l 1

G v V (G) G v A = [X 0,..., X r ] r = v (G) X 0 = {v} X i+1 = N G (X i )\ j=0,...,i 1 X j i = 1,..., r X 3 X 2 X 1 x X 0 x A = [X 0,..., X r ] G v i=0,...,r X i = V (G) A = [X 0,..., X r ] G v i, j, 0 i j r x, y x X i y X j P x y X i,..., X j P X i P [a 1,..., a q ] {0,..., r} a 1 = i a q = j a h, a h+1, 1 j < q a h a h+1 1 A X i X i 1 X i X i+1 {i,..., j} G A = [X 0,..., X r ] G v i = 0,..., r X i G i v i u X i G (v, u) = i i i = 0 i j i = j + 1 u X j+1 X j+1 u u X j j v u P G v u j + 1 P G v u j P X 0,..., X j+1 P j

A A G (v, u) = i u X h, h {1,..., i 1, i + 1,..., r} A = [X 0,..., X r ] V (G) u V (G)\ h {1,...,i 1,i+1,...,r} X h = X i G (G) d v V (G) 1 + ((d 1) 1) G l v d d 2 A = [X 0,..., X r ] G v G i v X i X i G v X i X i d (d 1) i 1 i 1 i = 1,..., l G i v i=0,...,l X i X i 1 + d + d(d 1) + + d(d 1) l 1 i=0,...,l = 1 + d( i=0,...,l 1 (d 1) i ) = 1 + d ((d 1) 1) d 2 G (G) α (G) d n(g) 1 + d d 2 ((d 1)α 1). v l = (G) G (G) v G (G) β (G) d n(g) 1 + d d 2 ((d 1)β 1). A = [X 0,..., X r ] G v { X i 0 i r} G v v G G (G) G (G) n(g) 1 (G) v G G G v A = [X 0,..., X r ] G v n(g) 1 + r X i 1 + r (G) r = G (v) (G) n(g) 1 + (G) (G)

G n (G) d d n/2 G n (G) d (G) β m(g) n(n 1)(d 2) 2((d 1) β 1). e G 2 l (d 1) l 1 l 1 Pi r, i = 1,..., r r e = (x, y) i (d 1)(d 1) r i 1 = (d 1) r i r i y G\e (d 1)(d 1) i 2 = (d 1) i 1 i 1 x G\e Pi r e x y (d 1) r i (d 1) i 1 = (d 1) r 1 e Pi r 2 (d 1)r 1 e l r G 2 m(g) (d 1) r 1 G 2 (n 2) G β 2 ( ) n 2 2 m(g) i=1,...,β (d 1) i 1 m(g) G n (G) d (G) β m(g) d n/2 d n 2 n(n 1)(d 2) 2((d 1) β 1). n G G (G) G G (G) G (G) = (G) = 0

G H G H 3 C = (v 1,..., v r, v 1 ) {z, y} z y C G (G) G δ(g) (G) 1 P = (v 1,..., v t ) G v 1 v 1 v i i G (v 1 ) + 1 δ(g) + 1 δ(g) + 1 G G ϵ(g) 1 V (H) 3 < n n = n(g) δ(g) 2 G v 1 ϵ(g\v) 1 G\v G ϵ(g) 1 K 3 τπ G K 3 G G (G) g δ(g) d { 1 + d i=0,...,r 1 n(g) (d 1)i g = 2r + 1 g 2 i=0,...,r 1 (d 1)i g = 2r g g 2 = 1 S i, 0 i r r + 1 G v 0 G i = 1,..., r v S i S i 1

G v v 0 i i v 0 2r < g S i (d 1) S i 1 2 i r S 0 = 1 S 1 d n(g) S i 1 + d + d(d 1) +..., d(d 1) r 1 i=0,...,r g 2 = 0 v 0 G n(g ) S i 1 + 2 + 2(d 1) +..., 2(d 1) r 1 = 1 + 2 (d 1) i i=0,...,r n(g) = n(g ) 1 i=0,...,r 1 G n n+n 1+ 1 k (G) 2k k n + 1 = d δ (G) ϵ(g) d G G δ(g ) d (G) 2k + 1 (G ) 2k + 1 d > 2 G n n(g ) 1 + d i=0,...,r 1 d > 2 (d 1) i = 1 + d d 2 ((d 1)k 1) > (d 1) k = n, k 1 {(H) H υπ P k P k } = k(k + 2). G G (p, q) p, q 1

r r 1 G H G (G H) G G (G) = (G) = V (G) x, y x y 2x x y G (G) = (G) G (G) < 3 (G) > 3 x, y x y 2x G (G) = x (G) = y G δ(g) (G) 2 G G G n (G) x n x G (G) 2 (G) + 1

ΚΕΦΑΛΑΙΟ 5 x, y V (G) (x, y) G (G) < K 1 G v G G (v) 1 G G\v [v 1,..., v n ] G i = 1,..., n 1 (v i, v i+1 ) P i G G P i (v i, v i+1 ) W i W 1,..., W n 1 G G G n(g) n(g) = 1 < n G n(g) = n v V (G) N G (v) = V (G)\{v} x, y V (G)\{v} {v, x} E(G) {v, y} E(G) H = G[V (G)\{v}] < n H H {v, x} G G {v, y} G G

G I(G) G H I(G) G I(G) H G G G H G δ(h) δ(g) (H) (G) G δ(g) n(g) 2 G G H n(h) n(g) 2 δ(h) n(h) 1 < n(g) 2 G m(g) n(g) 1 G G m(g) < n(g) 1 H n(h) < n(g) m(h) n(h) 1 δ(g) 1 m(g) n(g) δ(g) 2 n(g) v G G H = G\v G m(h) n(h) 1 m(g) = m(h) + 1 n(g) = n(h) + 1 m(g) n(g) 1 G S V (G) S G G\S S S G S S (a, b) a, b V (G) G\S (a, b) S (a, b) (a, b) S (a, b) G S k 2 G G G

a e i b f j G c d g h k l {e, f, g, h} G {e, f, h} {e, g} G {f} {h} G {f, g} (a, k) {h} (a, k) G G v x, y G\v x y v x y P 1 x t w y P 2 P 1 P 2 G x, y V (G) G (x, y) x y G G (x, y) = 1 e = {x, y} G e G G\x {x} G G\x (y) 1 G\e G\e G\x G\e G\e x y {x, y} G x y (x, y) < k x, y G (x, y) = k 2 w k G x y P 1 P 2 G x w P 1 P 1, P 2 y P P 1 x y {y, w} P, P 2 G P 1, P 2 y R

G x y w G G\w R P 1 P 2 R P 1 {w, y} G t P 1 P 2 R t P 1 P P 1 x t R t y P 2 P 2 {w, y} P 2 P G G 3 x, y, z V (G) G y x z G + G w x z G + P 1, P 2 w y (P 1 P 2 )\w H 1 H 2 V (H 1 ) V (H 2 ) 2 H 1 H 2 H 1 x 1 v u u v x 2 H 2 {v, u} S = V (H 1 ) V (H 2 ) x 1, x 2 H = H 1 H 2 x 1 x 2 H 1 H 2 x 1 V (H 1 )\S x 2 V (H 2 )\S P 1 G 1 v u x 1 P 1 P 1 v, u S P 2 G 2 v u x 1 P 1 P 2 v G {v} G

G I 2 (G) G H I 2 (G) G G K 2 I 2 (G) H 1 H 2 G {x, y} S = V (H 1 ) V (H 2 ) w V (H 1 )\V (H 2 ) P H 1 x y w H 1 P H 1 H 1 P H 1 x P x v P y x y P x y V (H 1 ) V (H 2 ) = {v} G\v x y H 1 H 2 v P x y G\v H = H 1 H 2 H\v x y P H\v E(G)\E(H) H + = H P H + H 1 H H + P P H 1 H 2 x y x y v P x P y x y v H 1 H 2 C = P P x P y H + = C H 1 H 2 K 3

K 3 G k > k k κ(g) = {k G k } G κ(g) δ(g) G e E(G) κ(g\e) κ(g) 1 v V (G) κ(g\v) κ(g) 1 G S V (G) x V (G)\S (x, S) S x S G s, t G (s, t) G (s, t) G G (s, t) S k k (s, t) G S (s, t) S G k k (s, t) G k = 1 k > 1 k k H H (s, t) S k G G H G k (s, t) G e E(G) (s, t) S e k 1 G\e e E(G) e S e = w e\{s,t} S e {w} (s, t) G k

G\S e (s, t) S e = k 1 k s t e G\e e S e S e e s t (s, t) G N G (s) N G (t) = x t s S = S {t,x} \{x} (s, t) G\x S = k 1 k 1 (s, t) G\x s, x, t k (s, t) G s S t s t s S t G s G t G (s, t) S k G N G (s) = S N G (t) = S S (s, t) k G P s G s S S P t (s, t) G S P s P t P P s P P t V (P ) V (P ) = V (P ) V (P ) = {q} q S (s, t) S G s = P Ps P G t = P Pt P S V (G s )\s S V (G t )\t G s = G s {S {t}, {{x, t} s S}} G t = G s {S {t}, {{s, x} x S}} n(g t ), n(g s ) < n(g) k (s, t) P 1 s,..., P k s P 1 t,..., P k t G s G t (s, S) {Q i s i = 1,..., k} = {P i s\t, i = 1,..., k} (t, S) {Q i t i = 1,..., k} = {P i t \s i = 1,..., k} Q 1 s Q 1 t,..., Q k s Q k t k (s, t) G P (s, t) G [s, v 1, v 2,..., t] e = {v 1, v 2 } v 2 t {v 1, t} E(G) P 3 {v 1 } S e (s, t) S k G {v 1, t} E(G) N G (s) = {v 1 } S e P {s, v 2 } E(G) {v 2 } S e (s, t) S k G {s, v 2 } E(G) N G (t) = {v 2 } S e k 2 S e s t (s, t) (s, t)

x y (x, y) G κ G (x, y) κ(g) = {κ G (x, y) x, y V (G), {x, y} E(G)} k k S (x, y) G G (x, S) W x (y, S) W y W x G\S x e k G κ(g\e) = k 1 G κ(g) = k e = {x, y} E(G) e G κ G\e (x, y) = k 1 G = G\e e G R V (G ) k 1 G \R x y G \R R G κ(g) = k R (x, y) G k 1 κ G (x, y) k 1 κ(g ) k 1 κ G (x, y) k 1 κ G (x, y) = k 1 e G k k (x, y) G G (x, y) G k κ(g\e)(x, y) k 1 G δ(g) > κ(g) e E(G) κ(g\e) = κ(g) G G k κ(g) = k G k S G\S v N G (v) = S G (v) = k C D = G\S\V (C) D G\S n(d) n(c) e = {x, y} x, y V (C) G = G\e G (x, y) R k 1

v G G = G [S C] ({v } S, {{v, w} w S}) k (v, x) G (v, x) S G k 1 S S V (C) S (z, x) G z D S + = S {y} (z, x) G S + S V (C) S + S C + G\S + x C S (x, S) W x G G W x (x, S) G (y, S) W x G D S C x z y S G z V (D) k (z, x) G (z, S) W z G G V (D) R z V (D)\R G W z W x k (x, z) G W z W y k (z, y) R z x z z y R x y V (D) R V (D) V (D) < V (C) R R 1 = R V (D) = V (D) R 2 = R S R 3 = R V (C) R (x, y) G w S\R R W x W y x w w y R R 3 1 2 S\R R 2 = S S\R R 3 1 2 ( S R 2 ) = 1 2 (k R 2 ) V (D) = R 1 = R R 3 R 2 = k 1 R 3 R 2 k R 2 1 2 (k R 2 ) 1 = 1 2 (k R 2 ) 1 < 1 2 (k R 2 ) R 3 V (C) r 0 K 2,r + = K 2 (r K 1 ) K 2 K 2,r + K 2,r

K + 2,5 G k k + 2 k 2 v V (G) d K + 2,d 2 πα G[N G (v)] G e E(G) κ(g\e) = k e = {x, y} G[N G (v)] K + 2,d 2 S = v N G (v)\{x, y} S 2 G = (G\S)\e P G S G\e k 1 (x, y) S P κ G (x, y) = k e G G v G G G K 3 2 G G G v 2 v G K 3 n(g) 4 G/v 2 r W r = C r K 1 W r r 3 3 e E(G) G\e 3 e E(G) G G/e 3 G G = K 4 K 4 W 3 n

W 9 n(g) = n G G G G G e = {a, b} G v e G e = (G\a)\b G W r v V (G) v 1, v 2, v 3 K 3 K 2,1 + v 1, v 2, v 3 G v 1, v 2, v 3 G v 1, v 2, v 3 G e = {v, v 3 } v e G e e v 1 v 2 B 1, B 2 G e v e {v 3, v e } G B G e {a, v 3 } G a B v 3 B 1 \v e B 2 \v e v 3 1 B 2 {v e, v 2 } {v e, v 1 } G v 3 w i B i \v e \v i i = 1, 2 {v i, v e } G i = 1 2 f = {v, v 1 } S f = {v 1, v f } G = G\v (α) {v, v 1, v i }, i = 2, 3 G v f {v 2, v 3 } (β) v 2, v 3 G \S f S f G G v 2 v 3 S f (v 2, v 3 ) G v 1 (α) {v 1, v 2 } v 1, v 2, v 3 G e = {v, v 3 } G S = {v, v 3, v e } C D G\S {v 1, v 2 } {v e } C v G S C v D {v 3, v e } G

v e f B v 1 1 w 1 v v 3 G v e B 2 v 2 S f v f v 1 v 3 v 2 w 2 G (α) (β) α G G e β S f G {v 1, v 2 } {v 1, v 3 } v 1, v 2, v 3 G G G\v {v 2, v 3 } G 3 S G S 2 v 1, v 2, v 3 G S C G\S S G G e = {v 2, v 3 } G f = {x, y} G (x, y) G \f (x, y) G {v 2, v 3 } {v 2, v} {v, v 3 } f G e = {v 2, v 3 } G H = G G \e H = G \e H n(g) < n(h) H f H/f f e e E(H) = G \e e H H = G f G G H W r r 3 = G \e G W r v a, b, c W r K 1,2 W r {a, b}, {b, c} E(W r ) {a, c} E(W r ) K 1,2 (α) (β) {a, b} {b, c} H = G G W r

a v a v b c b c (α) (β) a v a v c c b b (γ) (δ) W r+1 v W r W r (γ) W r W r (δ) W r+1 W 4 Q 3 K 4 Q 8

v 4 v 1 v 2 v v 1 v 2 v 1 v 2 3 G 5 e 3 H G E πα G e v e 3 H E E e v e G = (V (H ), V (H ) H ) G G = G/e G 3 G 1,..., G m G 1 = G G m = K 4 i = 1,... m 1 G i G i+1 G n(g) 4 κ(g) 3 K 4 ϵλ G G δ(g) 3 G G G V (G) 5 3 < 5 K 4 κ(k 4 ) = 3 κ(g) 2 S G C G\S C + = G[S V (C)] S = {x} x C + N C +(x) S G G\S C C δ(c + ) 3 n(c + ) < n(g) C + G S = {x, y} (x, y) P G\V (C) C C + {x, y} (x, y) P G\V (C) C ϵλ G S C N C (x) N C (y) G C δ(c ) 3 n(c ) < n(g) C G

G δ(g) 3 K 3 ϵλ G k κ (G) = {k G k } G m(g) (2 1)(n(G) k) k n(g) = 1 < n G n(g) = n m(g) (2 1)(n(G) k) S G S k G k S < k C 1 G\S G 1 = G[V (C 1 ) S] G 2 = G\V (C 1 ) S = V (G 1 ) V (G 2 ) G 1, G 2 m(g) m(g 1 ) + m(g 2 ) n(g 1 ) + n(g 2 ) = n(g) + S. h {1, 2} m(g h ) (2k 1)(n(G h ) k), (2 1)(n(G) k) m(g) m(g 1 )+m(g 2 ) < (2k 1)(n(G 1 )+n(g 2 ) 2k)) = (2k 1)(n(G)+ S 2k) < (2k 1)(n(G) k)), n(g i ) < n, i = 1, 2 G h k G h G G H κ(h) ϵ(g) 2 κ (G) ϵ(g) 2 ϵ(g) 2 k κ (G) k ϵ(g) 2 k m(g) 2k n(g) (2 1)(n(G) k) G k κ (G) k G G λ(g) G λ(g) = { F F E(F ) G\F }.

G k K k G G n V (G) = {v 1,..., v n } G[{v 1,..., v i }] i = 1,..., n G G G K 2,3 ϵλ G G K 1 K 2 G a b b a = (t(v) 1) v V (G) t(v) v 4 n n 1

Q 8 K 4 G 2κ (G) ϵ(g) κ(g) 2 δ(g) (n(g) + k 2)/2 κ(g) k f : NN k N G δ(g) f(k) k ϵ (G) = {k H υπ G : ϵ(h) k}. ϵ (G) δ (G) 2 ϵ (G) κ (G) δ (G) 4 κ (G) ϵ (G) 2 κ (G) 4 ϵ (G)

ΚΕΦΑΛΑΙΟ 6 G x, y V (G) P 1 P 2 e = {x, y} P 1 P 2 H = (P 1 P 2 )\e G H P x y H P G\e P ({x, y}, {{x, y}}) G G G < n G n x y G G (x) = 1 G\x x G G m(g) = n(g) 1 G m(g) n(g) 1 m(g) n(g) 1 m(g) n(g) ϵ(g) 1 G n(t ) 2

2 m(t ) 1 + 2(n(G) 1) m(t ) n(g) 1 2 m(t ) n(g) G δ(g) + 1 δ(g) δ(g) = 1 2 δ(g) = k 1 k 1 k G k T k + 1 T G T = T \v v T G = G\y y G δ(g ) k 1 n(t ) = k T G σ : V (T ) V (G ) T G u T v u T u = σ(u) G T G T σ v G u k 1 G (u ) k u x G T V (T )\{u } = k 1 σ σ(v) = x T G G (u ) = k 1 u G y u G k 1 σ σ(v) = y T G G G G G G G G G G G T V (T ) V (G) v V (G)\V (T ) V (T ) 1 G

u V (T ) P (v, u) G x 0 = v, x 1,..., x r = u G v V (T ) {x 0, x 1 } T P e i = {x i, x i+1 }, i 1 P T e T G G G T G G G n(g) = m(g) 1 G T n(t ) 1 G T G = T n n {1,..., n} (T, τ) T τ : V (T ) {1,..., n(t )} V (T ) n(g) n(t ) (T, τ) (T, τ ) σ : V (T ) V (T ) T T v V (T ), τ(v) = τ (σ(v)) n n 2 n n 1 n n 2 n n n n 2 n 2 n n A = (a 1,..., a n 2 ) n

S = {1,..., n} T = (V, E) V n E = τ : V S V S S > 2 x S A x S y A E {τ 1 (x), τ 1 (y)} A S (T, τ) (T, τ) n() A V (T ) > 2 v T w v A τ(w) T v A 10 8 9 1 T 5 2 3 4 6 7 [5, 5, 2, 3, 3, 2, 8, 8] A (T, τ) (T, τ) A n n 2 n

3 T 1 T 2 δ (T 1 T 2 ) 3 G n(g) m(g) G m(g) n(g) 1 (T ) G n(g) m(g) T G δ(g) n(t ) 1 T G k

ΚΕΦΑΛΑΙΟ 7 R 2 S R 2 S S S Γ = (V, A) v V R 2 Γ e A R 2 (0, 1) e e e V V ( e E e) = Γ = V ( e E e) E V (Γ) = V E(Γ) = A

f 3 f 3 f 1 f 2 f 1 f 2 f 4 f 4 f 5 f 5 Γ R 2 Γ F (Γ) Γ K 5 K 3,3 Γ = (V, E) Γ = (V, E ) Γ V V E E (V, E ) R 2 \ Γ Γ = (V, E) R 2 \ Γ Γ F (Γ) Γ G Γ = (V, E) D R 2 Γ D R 2 {(x, y) R 2 x 2 + y 2 < 1} Γ D Γ = (V, E) f F (Γ) Γ Γ Γ Γ = (V, E) G Γ = (V, { e e E(Γ)}). Γ G Γ Γ f F (Γ) Γ[f] Γ V (Γ) f f\v (Γ) f F (G) G Γ[f] K 3 Γ f 1 f 4 G Γ G G Γ G) G Γ G Γ G Γ G

G Γ Γ υπ, ϵν, πα, τπ ϵλ G Γ Π Γ Π G Γ Π Γ G Γ G H G H K r, r 2 G H Γ e Γ f 1, f 2 F (Γ) e f 1 f 2 Γ f F (G) Γ[f] Γ Λ Γ Γ Γ Γ R 2 \ Λ Λ Γ

W 1 W 2 G W 1 W 2 [v 1, v 2, v 3, v 1, v 5, v 1, v 4, v 1 ] [v 4, v 1, v 5, v 1, v 3, v 2, v 1, v 4 ] [v 4, v 1, v 5, v 1, v 2, v 3, v 1, v 4 ] Γ = (V, A) f F (Γ) f Γ V f f\v j i k f 4 g a l b f 2 f 3 c h f 1 e G π(f 1 ) = [e, h, c, b, a, j, c, j, i, c, h, g, e], π(f 2 ) = [b, a, k, l, k, c, b], π(f 3 ) = [g, e, h, g], π(f 4 ) = [i, c, j, i] Γ = (V, E) f F (Γ) π(f) f Γ = (V, E) Γ = (V, E) Γ Γ G Γ G Γ ρ : V (Γ) V (Γ ) σ : F (Γ) F (Γ ) f F (Γ) ρ(π(f)) = π(σ(f)) Γ Γ G Γ G Γ G G Γ, Γ G G Γ Γ

σ(f 1 ) f 1 f 3 σ(f 2 ) σ(f 3 ) f 2 Γ Γ Γ Γ Γ G Γ G Γ Γ Γ R 2 S 0 = {(x, y, z) x 2 +y 2 +(z 1) 2 = 1} (0, 0, 2) (x, y, z) S 0 = {(x, y, z) x 2 + y 2 + (z 1) 2 = 1} (χ, ψ) R 2 x (x, y) ( 2 z, y 2 z ) 2x (x, y, z) ( x 2 + y 2 + 1, 2y x 2 + y 2 + 1, 2x 2 + 2y 2 x 2 + y 2 + 1 ) f Γ s = (x 0, y 0 ) f Γ

(x, y) (x x 0, y y 0 ) s = (x 0, y 0 )) G Γ S 0 s (x, y, z) (2 x, 2 y, z) {(x, y, z) R 3 z = 0} s (0, 0, 2) s G Γ Γ Γ f Γ Γ f Γ Γ ρ π f π(f) Γ Γ = (V, A) F = F (Γ) Γ = (V, A ) Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Γ Q 3 K 2,2,2

Γ Γ Γ Γ G H G K 2,2,2 K [3] 2 G K 4 Γ n m r m + 2 = r + n Γ m n 1 m = n 1 G Γ r = 1

< m n Γ n m r m n G Γ e Γ e f, f Γ e Γ Γ = (V (Γ), E(Γ)\{e}) f f f f e Γ (m 1) + 2 = (r 1) + n Γ V (G) 3 Γ E(Γ) Γ Γ Γ V (Γ) 3 Γ Γ 3 3n 6 Γ 2 3 E(Γ) r = 2 3m G n 3 3n 6 G δ(g) 5 m = m(g) n = n(g) Γ 6 m 6 2 n = 3 n G δ (G) 5 n 3 n

G G x G y x, y 3 n = n(g) r = n(g ) n, r, x y n + r = xn + 2, 2 n + r = yr + 2, 2 x, y 3, x, y 5, n = 4y 2(x + y) xy x = 3 y = 3 n = 4 r = 4 G x = 3 y = 4 n = 8 r = 6 G x = 4 y = 3 n = 6 r = 8 G x = 3 y = 5 n = 12 r = 20 G x = 5 y = 3 n = 20 r = 12 G x, y 4 (x 4)(y 4) 0 xy 4x 4y 16 0 xy 2(x + y) 2(x 4) + 2(y 4) 0

K 5 K 3,3 G K 3,3 < m G m G e = {x, y} E(G) G G = G e G e = {x, y} E(G) G G = G/e v e e G = G\e G = G/e m(g ) < m(g) G K 5 K 3,3 Γ G x y Γ e Γ G Γ C x y R\ C S = V (S) G G 3 (x, y) P i, i = 1, 2, 3 G {x, y} E(G ) S = V (S) F = C P 1 P 2 P 3 K5 = (2 K 1) K 3 {x, y} F K 5 K 5 τπ G (α) Γ Γ = Γ \v e f F (Γ ) Γ \v e v e Γ Γ [f] C M = [x 1,..., x r, x 1 ] S e = {x, y} G X = N G (x)\y Y = N G (y)\x V (C) X M Y G K 3,3 τπ G (β) K 5 K 3,3 K 5 K 3,3 G K 5 K 3,3 G S S = 1 2 C i, i = 1,..., r G\S i = 1,..., r G i G V (C i ) S i = 1,..., r G i ϵλ G n(g i ) < n(g) G G i K 5 K 3,3 G K 5 K 3,3 G i, i = 1,..., r i, j 1 i < j r G i G j = K S i=1,...,r G i = G

G e E(G) K 5 τπ G/e K 5 τπ G K 3,3 τπ G A xy A x A y x y K 3,3 e = {x, y} v e G = G/e e A xy = N G (x) N G (y) A x = N G (x)\{y}\a xy, A x = N G (y)\{x}\a xy H G D V (G ) v e D K 5 τπ G v e V (H)\D N = N H (v e ) N = 4 N A x, A y, A xy N A x A y K 3,3 τπ G K 5 τπ G G K 5 K 3,3 K 5 K 3,3 K 5 K 3,3 Γ υπ, ϵν, πα, τπ ϵλ K 4 K 2,3

K 4 K 2,3 K 5 K 3,3 K 4 K 2,3 G G + = G K 1 G G G + G + Γ Γ Γ G G + G + K 5 K 3,3 G K 4 K 2,3 G n 2n 3 v 1 v 5 Γ v 2 v 3 v 4 v 2 v 3 v 4 Γ v 5 v 1 G Γ G f Γ Γ[f] [v 1,..., v n, v 1 ] Γ + f Γ Γ [v 1,..., v n, v 1 ] Γ Γ + {v i, v n i+1 } i = 1,..., r {v i, v n i } i = 1,..., r 1 {v n, v n} G

n(g ) = 2 n(g) m(g ) = 2 n(g) + 2 m(g)) m(g ) 3 n(g ) 6 2 n(g) + 2 m(g)) 6 n(g) 6 m(g) 2 n(g) 3 G (G) 3 G H G H G (G) 3 r 3 r = 3 r = 4 ξ P X,Y = {(x, y, z) R 3 z = 0} R 3 P X,Y S 0 ξ G G H m

n 2(n 1) 6 4 G K 3 υπ G δ (G) 3 C 4 τπ G m(g) 3 2 (n 1) Γ κ r n m m + κ + 1 = n + r 6 x δ(g) 2 δ() 2 G H H G 6 H G δ (G) 6 K 4 τπ G 1 2 (3n 1) 4 C 4 τπ G m(g) 3 2 (n 1) n 0 n

H 3 n K4 K 4 G = {G K4 ϵλ G} {V 1, V 2 } V (G) G[V 1 ], G[V 2 ]

ΚΕΦΑΛΑΙΟ 8 k k k G χ : V (G) {1,..., k} {x, y} E(G) χ(v) χ(u) k G G k k χ 1 (i) i = 1,..., k χ S V (G) χ(s) = {χ(v) v S} X {1,..., k} χ 1 (S) = {χ 1 (i) i X} χ k G k G k χ(g) k 2 χ(c 2k 1 ) = 3 χ(c 2k ) = 2 l 1,..., l k K l1,...,l k = K l1 + + K lk V i (K l1,...,l k ) = V (K li ), i = 1,..., k K l1,...,l k k k K l1,...,l k l 1,..., l k V i (G), i = 1,..., k k G k

K K 3,3,3,3 k k k k k k χ : V (G) {1,..., k} k G G K χ 1 (1),..., χ 1 (k) χ : V (K l1,...,l k ) {1,..., k} χ(v) K l1,...,l k v χ K l1,...,l k G k G k G k G n n 2 ( k 1 2k ) G K l1,...,l k l 1,..., l k i=1,...,k l i = n m(g) m(k l1,...,l k ) K l1,...,l k m(k l1,...,l k ) ( ) n 2 i=1,...,k = 1 2 (n2 n = 1 2 (n2 i=1,...,k 1 2 (n2 n2 k ) = n 2 ( k 1 2k ) ( ) li 2 i=1,...,k (l 2 i )) (l 2 i l i ) i=1,...,k l2 i 1 k ( i=1,...,k l i) 2 k G n m χ(g) n2 n 2 2m

G 2 G G G G G G A = [X 0,..., X r ] G v G G X i {x, y} X i P x v x P x v y X 1,..., X i 1 w P 1 P 2 v P 1 P 2 P 1 P 2 w P 1 P 2 X i X i G n n2 4 G k S V (G) χ : V (G) {1,..., k} G χ(s) = {1,..., k} k G S S k G v, u S i j χ(v) = i, χ(u) = j G[χ 1 (i) χ 1 (j)] χ : V (G) {1,..., k} k G V v G[χ 1 (i) χ 1 (j)] v u V v χ G χ i j V v χ = χ\{(x, χ(x)) x V v } {(x, i + j χ(x)) x V v }. χ (v) = χ (u) = j χ 1 (S) = {1,..., k} i S

G χ(g) δ (G) + 1 G δ (G)+1 n(g) n(g) = 1 G < n δ (G) + 1 G n(g) = n v G δ (G) δ G\v δ (G\v) + 1 δ (G) + 1 χ : V (G\v) {1,..., δ (G) + 1} X = χ 1 (N G (v)) X δ (G) R = {1,..., δ (G) + 1}\X i R χ = χ {(v, i)} χ G δ (G) + 1 G l (l + 1) χ(g) l + 2 δ (G) l + 1 H H δ(h) l + 1 l + 2 l + 1 G n χ(g) + χ(g) n + 1 n χ(g) χ(g) χ(g) + χ(g) δ (G) + 1 + δ (G) + 1 δ (G) n δ (G) 1 G χ(g) K l1,...,l χ(g) l i = {l 1,..., l χ(g) } l i n χ(g) V i(g) G G G n χ(g) 6 5

5 G v 5 G G = G\v 5 S = N v (G) 5 χ G {1, 2, 3, 4, 5}\χ 1 (S) i χ {(v, i)} 5 G Γ G Γ N G (v) v [v 1, v 2, v 3, v 4, v 5, v 1 ] χ(v i ) = i i = 1,..., 5 i, j, 1 i < j 5 G i,j = G[χ 1 (i) χ 1 (j)] G i,j G i j v i v j G i,j i, j, 1 i < j 5 P v 1 v 2 v v 5 v 4 v 3 v 1 v 3 P G P v L G Λ Γ G Λ = L Λ R 2 R 1, R 2 R 2 \ˆΛ v 2, v 4 Γ v 2 v 4 Λ G 2,4 v 2 v 4 G 2,4 i = 2 j = 4 4 δ (G) (G)

G ( (G) + 1) (G) d 3 G (G) d K d+1 υπ G G d G v G G = G\v d S = N v (G) d χ G {1,..., d} \χ 1 (S) i χ {(v, i)} d G S S = {v 1,..., v d } χ(v i ) = i G v 1 v 2 G G i = 1,..., d S i = {v i } N G (v i ) h {1,..., i 1, i+1,..., d}\χ(s i ) = χ = χ\{(v i, χ(v i ))} {(v i, h)} G S G G i,j = G[χ 1 (i) χ 1 (j)] v i v j G i,j, P i,j (v i, v j ) G i,j i, j 1 i < j d P i,j = G i,j i, j 1 i < j d P i,j G i,j D = [v i, a 1,..., a p, v j ] P i,j P i,j v i b a 1 χ(b) = j S i Gi,j (v i ) = 1 Gi,j (v j ) = 1 a s D Gi,j (c) > 2 χ(a s ) = i d a s P i,j χ(d) = j a s j a s P i,j d {1,..., i 1, i + 1,..., d}\χ(n Gi (a s )) h χ = χ\{(a s, i)} {(a s, h)} d G d a h C i,j = C i,j\a h i j S i, j, k {1,..., d} V (C i,k ) V (C k,j ) = {x k } c V (C i,k ) V (C k,j )\{x k } χ(c) = k c i j {1,..., k 1, k + 1,..., d}\χ(n G (c)) h χ = χ\{(c, k)} {(c, h)} d G d C i,k = C i,k\c i k S z P 1,2 v 1 χ(z) = 2 z S z P 2,3 P 1,2 P 2,3

(G) G 2l G l 2 2 2 µ µ U n {v 1,..., v n } U n X m,n,d,k = {G U n m(g) = m nd, (G) d 2, χ(g) k} G m,n,d,l = {G U n m(g) = m nd, (G) d 2, (G) l}. n, d, k, l X dn,n,d,k < G dn,n,d,l U n l k X m,n,d,k G m,n,d,k X m,n,d,k χ : V (G) {1,..., k} ) k ( n/k ) = 1 2 n2 (1 1 k H = (V (G), ) ( n 2 2 ) χ ( 1 2 n2 (1 1 k ) ) m ( 1 2 n2 (1 1 k ))m m H χ k n k H X m,n,d,k k n ( 1 2 n2 (1 1 k ))m G m,n,d,l G m,n,d,l H G m 1,n,d,l e () d 2 m(h) nd 2n/d H d 2 n(1 2 d ) S V (H) e x S S\{x}

l v l d2 d 2 2 (d2 1) l d 2l n(1 2 d ) d2l e 1 2 n(1 2 d )(n(1 2 d ) d2l ) H G m,n,d,l G m,n,d,l ( 1 2 n(1 2 d )(n(1 2 d ) d2l )) m n d X dn,n,d,k G dn,n,d,l k n ( 1 2 n2 (1 1 k ))dn ( 1 2 n(1 2 d )(n(1 2 d ) d2l )) nd n 2 k 1/d (1 1 k ) n(n(1 2 d ) d2l )(1 2 d ). n d n 2 n 2 d k 1/d (1 1 k ) (1 2 d )2. d 1 1 1/k G K 5 K 3,3 K 5 K 3 4 K 5 G K 5 ϵλ G V 8 G K 5 ϵλ G

V 8 V 8 G K 4 ϵλ G k 0 G K k+1 ϵλ G k k = 6 r 7 K r G K r ϵλ G ϵ(g) 2 r 2 c(r) c ϵ(g) c K t ϵλ G c(t) = (α + o(1))t t α = 0.3190863... r = 1, 2, 3 r = 4 G 2007 3 2006 G χ(g) 1 2 + 2m(G) + 1 4

G {V 1, V 2 } V (G) χ(g[v 1 ])+ χ(g[v 2 ]) = χ(g) G {V 1, V 2 } V (G) χ(g[v 1 ]) + χ(g[v 2 ]) > χ(g) G H χ(g 1 ) χ(g 2 ) χ(g 1 G 2 ) G 89 2007 G l (l + 1)

ΚΕΦΑΛΑΙΟ 9 ω(g) G G ω(g) = {k K k υπ G} G G ω(g) χ(g) G ω(g) 4 τ(p, n) p n p, n (r 1, n 0) p n 1,..., n p G m(g) = n i n j 1 i<j p n 1,..., n p n/p p

n p,..., n p, n p,..., n p. } {{ } } {{ } n p p (n p) p, n (r 1, n 0) T p (n) τ(p, n) T 4 (10) T 5 (9) G (k, ω) ω(g) k G ω(g ) k n(g) = n(g ) m(g) < m(g ) G v V (G) v G G v G N G (v) v v v v G G + G ω(g) = ω(g + ) (k, ω) G x, y, a {x, y} E(G) {x, a}, {y, a} E(G) x y

G (x) > G (a) x G + ω(g + ) k m(g + ) = m(g) + G (x) G + a G ω(g ) k m(g ) = m(g) + G (x) G (a) > m(g) G (x) G (a) G (y) G (a) a G + ω(g + ) k m(g + ) = m(g)+2 G (a) G + x y G ω(g ) k m(g ) = m(g)+2 G (a) G (x) ( G (y) 1) > m(g) x y x y G (y) 1 (k, ω) G n T k (n) G k ω(g) > k G T k (n) (k, ω) G m(g) τ(ω(g), n(g)) G S V (G) G S G α(g) G G G α(g) = ω(g) G n(g) α(g) χ(g) k l r(k, l) k l n G n ω(g) k α(g) l k l r(1, l) = r(k, 1) = 1 r(2, l) = r r(k, 2) = k r(k, l) = r(l, k)

r(k, l) k l r(k, l) r(k 1, l) + r(k, l 1). G G n(g) r(k 1, l) + r(k, l 1) v G k 1 = N G (v) k 2 = N G (v) k 2 G v G k 1 + k 2 = n(g) 1 r(k 1, l) + r(k, l 1) 1. k 1 r(k 1, l) G = G[N G (v)] ω(g ) k 1 α(g ) l ω(g) k v G + α(g) l k 1 < r(k 1, l) k 1 r(k 1, l) 1 k 1 r(k 1, l) + 1 k 2 r(k, l 1) G ω(g ) k α(g ) l 1 ω(g) k α(g) l v G k l ( ) k + l 2 r(k, l). k 1 k + l k +l 5 p, q k, l k + l < p + q r(p, q) r(p 1, q) + r(p, p 1) ( ) ( ) p + q 3 p + q 3 + p 1 p 2 ( ) p + q 2 =, p 1 r(3, 3) r(2, 3+r(3, 2)) = 6 ω(c 5 ) = α(c 5 ) = 2 r(3, 3) 6 r(3, 3) = 6 r(k, l) k l r(3, i) i {3,..., 9} r(4, i) i {4, 5} r(5, 5) {43,..., 49} r(5, 5) r(6, 6) r(5, 5) r(6, 6)

k r(k, k) 2 k/2 k 3 V n = {v 1,..., v n } G n V n G k n G n k i, j, 1 i < j n G n G n = 2 (n 2) S V n k 2 (n 2) ( k 2) Gn S ( n ) k S G k n ( ) n 2 (n 2) ( k G 2) n k k G n ( n )2 (k2) n k 2 (k 2) <. k k! n < 2 k/2 Gn k G n < 2k2/2 2 ( k 2) k! = 2k/2 k! < 1 2. G n k G n = {G G G n } G n k G G n } k ω(g) < k α(g) < k r(k, k) < 2 k/2 G ω(g) < k α(g) < k n < 2 k/2 G ω(g) δ (G) + 1 t(p, n) t(p, n) n 2 p 1 2p n p n p

ΚΕΦΑΛΑΙΟ 10 e G S V (G) S S V (G) G S G (G) G G S S G k G U D (G) { U, D } G (G) = n(g) α(g) S G G S G S V (G)\S S G

G S V (G)\S V (G)\S G G k n(g) k G δ (G) (G) δ (G) k G δ(h) k H S S < k H\S I v I H S G (v) S < k δ(h) k (G) (H) k G L(G) L(G) G L(G) L(G) G (G) = χ(l(g)) r L(G) G V (L) r L(G) G G r E(G) r G r r G r L(G) G M E(G) e,e M e e = µ(g) G M v V (G) v M G µ(g) = ω(l(g)) G χ(g) n(g) µ(g)

n = n(g) = n(g) µ(g) G G n 2 µ(g) G M G n 2 µ(g) µ(g) + n 2 µ(g) = n µ(g) χ(g) n(g) µ(g) G n m µ(g) 2mn n + 2m. µ(g) n χ(g) χ(g) n 2 n 2 2(( n 2) m) µ(g) n n 2 n 2 n(n 1)+2m 2mn n+2m 3 K 2 G µ(g) (G) G (G) = µ(g) G (G) µ(g) U D G M G M U U G S U M P S S M P M R G e M e = {u, d} u U d D d R S d R u R = M R G e E(G) R e M e M M e = {u, d} e M M {e} d e u S e S R d R e R u e = {u, d } S d R u R e R S P d d d e M d R e R d P e P P M e P d P P P + = P ({d, u, d, {e, e})} S d d R d e M P + M +

U u u U u e e e e D d D d d U u U u D e e d e d D e e d d M P + M P + M + G M G U D M U R U N G (R) R M U U S U M M M S D N G (S) M = S M N G (S) S M U (G) = µ(g) < U S G < U S U = S U S D = S D S G (U\S U ) (D\S D ) G N G (U\S U ) S D S < U S\S U < U\S U N G (U\S U ) S D = S\S U < U\S U R = U\S U N G (R) < R n m (G) m n α(g) n2 m n

ΚΕΦΑΛΑΙΟ 11 G H χ(g) = ω(g) 3

5 W i, i 4 i C 5 G G n χ(g) ω(g) χ(g) n µ(g) µ(g) = (G) ω(g) = α(g) α(g) = n (G) χ(g) n µ(g) = n (G) = α(g) = ω(g) H L(H) H H χ(l(h)) = ω(l(h)) (H) = χ(l(h)) µ(h) = ω(l(h)) G (G) = 3 G χ(g) = ω(g) = δ (G) + 1 = 4

S G G x, y S {x, y} E(G) S (a, b) a, b G\S C a C b G\S a b x y C a C b S (a, b) G (x, y) P a C a (x, y) P b C b P a P b P a P b G 4 G G G G a b S (a, b) G S G C a G\S a G 1 = G[S C a ] G 2 = G\C a G 1 G 2 n(g i ) < n(g), i = 1, 2 i = 1, 2 G i i {1, 2} G i v i V (G i )\S G i {1, 2} G i v i S v 1 v 2 G G δ (H) ω(g) 1 G ω(g) 1 G v G = G[N G (v)] G (v) = G (v) ω(g) 1 G H χ(h) δ (H) + 1 ω(h) 1 + 1 = ω(h) G {V c, V d } V (G) V c G V d G G H G (G) = {ω(h) 1 G H H }. G

C i G {V c, V d } V (G) G[V c ] C i V c G G[V d ] i 3 V d G i 3 0 i 3 G G I = {I 1,..., I n } I I i = [l i, r i ] l i < r i I G I = (I, {{I i, I j } I i I j }), G I I G I G I G I I G I G C i G i 4 I 1, I 2, I 3,..., I i C i I 1 l 1 = {l i 1 i i} I 3 r 1 I 1 I 3 G j = 1,..., i 2 I j+2 r j i 1 I 1 I i = I 1 I i G G

ω(g) = α(g) n(g) = n(g) n(h) α(h) ω(h) 2 3 2 2 3 1 4 1 4 1 4 5 G G 5 1 G 5 G I 0 = {3, 5} G 3 G 5 χ 3 χ 5 G 3 G 5 χ 3 χ 5 I 1 = {2, 5}, I 2 = {1, 4}, I 3 = {2, 4} I 4 = {1, 3} G G I 0, I 1, I 2, I 3, I 4 S 0 = {1, 2}, S 1 = {3, 4}, S 2 = {2, 3}, S 3 = {1, 5} S 4 = {4, 5} G H χ(h) = ω(h) n(h) α(h) χ(h) n(h) α(h) ω(h) G χ(g) > ω(g) H G G χ(h) = ω(h) p = ω(g) I 0 = {v 1,..., v q } G q = α(g) i {1,..., q} G i = G\v i ω(g i ) = ω(g) ω(g i ) < ω(g) χ(g i ) = ω(g i ) < ω(g) χ(g) ω(g) χ(g i ) = ω(g i ) = p i 1,..., q p σ i : V (G i ) {1,..., p} G i I (i 1) p+1,..., I i+p σ i i = 1,..., q G pq + 1 I 0, I 1,..., I pq G j {0,..., pq} χ(g\i j ) < ω(g) I j

χ(g) ω(g) χ(g\i j ) ω(g) ω(g) χ(g\i j ) = ω(g\i j ) ω(g) G\I j p S j pq + 1 S 0, S 1,..., S pq G j, j {0,..., pq} j j S j I j j = 0 j {1,..., pq} G i = G\v i σ i I j S 0 G\I 0 S 0 0 v i S 0 G[S 0 ] G i χ(g i ) = p G[S 0 ] σ i I j G[S 0 ] < p G[S 0 ] I j j {1,..., pq} j > 0 G[S j ] G\I j I j σ i i {1,..., q} σ i G i = G\v i v i S j v i S j G[S j ] G i \I j G i χ(g i \I j ) < p G[S j ] < p v i S j S j I 0 S j I j j {1,..., pq}\{j} G i = G\v i σ i I j i i v i S j v i, v i I 0 v i S j G[S j ] G i = G\v i G i I j S j < p i = i I j I j σ i S j I j S j I j = G[S j ] G\I j \I j G\{v i }\I j \I j p 2 G\I j \I j p 1 χ(s j ) < p 1 2 3 4 5 S 0 S 1 S 2 S 3 S 4 I 0 0 0 1 0 1 1 1 0 0 1 0 0 1 1 1 1 I 1 0 1 0 0 1 2 1 0 1 0 0 1 0 1 1 1 I 2 1 0 0 1 0 3 0 1 1 0 0 1 1 0 1 1 I 3 0 1 0 1 0 4 0 1 0 0 1 1 1 1 0 1 I 4 1 0 1 0 0 5 0 0 0 1 1 1 1 1 1 0 X Y Z X Y C 5 X, Y Z n(g) = 5 > 2 2 = α(g) ω(g) V (G) = {v 1,..., v n } (pq + 1) n X = [x i,h ] (i,j) [pq+1] [n] x i,h = 1 v j I i X I i, i = {0,..., pq} n (pq + 1) Y = [y h,j ] (h,j) [n] [pq+1] a h,j = 1 v h S j Y I j, j = {0,..., pq} S j I i i j i = j S j I i = 0 S j G\I j S j I j = S j I i = i j

S j I i S j I i S j I i = 1 z i,j = x i,h y h,j = S i I j h {1,...,n} XY (pq + 1) (pq + 1) Z = [z i,j ] (i,j) [pq+1] 2 Z 0 X n X n X pq Z = XY pq XY X Z Z pq + 1 P = (S, <) S R S x, y R x < y y < x R S x, y R x y y x a a a b c b c b c d e f d e f d e f g h g h g h P P P = (S, <) G P G G = (P, {{x, y}} x < y x > y}), S G G G P P = (S, <) G P P = (S, <) P = (S, <) S ρ P ρ P ρ

a b c d e f g h G P P = (S, <) S = n P B U D U = S v S v D (v, u) S S v < u B v U u D u D v U v D v v P a R U a b c d e f g h b c B d e f R D g h a b c d e f g h B B R {a, c, f, h} {b, d} {e, g} d e f P M B R B M = µ(b) = (B) = R = k R k S n k S P F P S F = E M {v, u } E A F v A u A F u

A v F [v, u] E E\{{v, u }} E = F n k U M M F = n k ρ = n k P = (S, <) P (P) F = {L 1,..., L k } S L i F P P = (S, <) P P F P ρ I ρ F P I F I = F F F P F I F = I I P = (S, <) S α(g P ) = (P) P G P α(g P ) P P P = (S, <) S (P) = χ(g P ) P G P F P ρ V (G P ) ρ G χ(g) ρ P (P) = χ(g P ) G n χ(g) ω(g) χ(g) ω(g) P = (V (G), <) G G = G P χ(g) = χ(g P ) = (P) = α(g P ) = α(g) = ω(g)

a a b c b c d e f d e f g h g h G P P G P G P G P G D (x, y), (y, z) E(D) (x, z) E(D) G (G) = ω(g) 1

ΚΕΦΑΛΑΙΟ 12 G G W = [v 0,..., v r 1, v 0 ] G W = [v 1,..., v r, v 1 ] e E(G) {i {v i, v i+1 r } = e} = 1.

G C = [v 0,..., v r 1, v 0 ] v G I {0,..., r 1} v = v i I = {i {0,..., r 1} v = v i } i I v = v i {v i 1 r, v i } {v i, v i+1 r } I C (v) = 2 I v ρ(g) = v V (G) ((v) 2). ρ(g) = 0 G G ρ(g) ρ(g) > 0 v 4 G v G {x, v} {y, v} v G w x y v G G G G (v) = G (v) 2 G G G w G v ρ(g ) < ρ(g) G G G G W = [v 0,..., v r 1 ] G

C A B D A B C D G G v 0 v r 1 G G G v 0 v r 1 G G G G G

G G G G G G I = {1,..., k} I 1, I 2 I I 1 + I 2 k + 2 i I 1 i + 1 I 2 j I 2 j + 1 I 1 G n(g) G x y G (x) + G (y) n(g) 2 < n(g) G P = [v 1,..., v r ] G r n(g) G P {v 1, v r } E(G) v 1 v r N G (v 1 ), N G (v 2 ) {v 2,..., v r 1 } N G (v 1 ) N G (v 2 ) r 2 n(g) 2 N G (v 1 ) + N G (v r ) n(g) i {2,..., r 2} v i N G (v r ) v i+1 N G (v 1 ) G C = [v 0, v i+1, v i+2,..., v r, v i, v i 1,..., v 0 ]

G r < n G w G C v C G w v C r + 1 P n(g)/2 G α(g) κ(g) C G G u V (G)\V (C) V (C) κ(g) V (C) < κ(g) x x C κ(g) G x x e = {x, x } V (C)\{x, x } < κ(g) 2 P V (C)\{x, x } P (C\e) r r κ(g) G + G v C V (C) κ(g) κ(g + ) κ(g) v u G + κ(g) v u G + (u, S) G S V (C) S κ(g) S u V (C)\S x S x C S C\{x, x } P 1 P 2 P 1, P 2 u x x C I V (C) S I G x, y I e C\{{x, x }, {y, y }} (e, {e}) P x P y x y x y P x P y u x y I V (C)\S u V (C)\S {u} I G κ(g) + 1 α(g) > κ(g)

n

k 3