W.G. Brown. On graphs that do not contain a Thomsen graph. Canad. Math. Bull., 9:281--285, 1966.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....follows from the fact that Theorem 1 was tight. 2 Theorems 2 and 3 can be proved similarly. It is enough to notice that splitting a vertex of high degree does not decrease the girth of a graph G and does not create a subgraph isomorphic to K r;s . Instead of Claim C, now we need . BS74] [B66], Be66] S66] W91] For a xed positive integer r, let G 2r denote the property that the girth of a graph is larger than 2r. Then the maximum number of edges of a graph with n vertices, which has property G 2r , satis es ex(n; G 2r ) O(n 1 1=r For r = 2; 3 and 5, this bound is tight. ....

....a xed positive integer r, let G 2r denote the property that the girth of a graph is larger than 2r. Then the maximum number of edges of a graph with n vertices, which has property G 2r , satis es ex(n; G 2r ) O(n 1 1=r For r = 2; 3 and 5, this bound is tight. KST54] F96] ER62] [B66], ARS98] For any integers s r 2, the maximum number of edges of a K r;s free graph of n vertices, satis es ex(n; K r;s ) O(n 2 1=r This bound is tight for s (r 1) In case r = 3, we obtain the following slight generalization of Theorem 2. Theorem 3.2. Let G be a graph of n vertices ....

W. G. Brown, On graphs that do not contain a Thomsen graph, Canadian Mathematical Bulletin 9 (1966), 281-285.

....follows from the fact that Theorem 1 was tight. # Theorems 2 and 3 can be proved similarly. It is enough to notice that splitting a vertex of high degree does not decrease the girth of a graph G and does not create a subgraph isomorphic to K r,s . Instead of Claim C, now we need . BS74] [B66], Be66] S66] W91] For a fixed positive integer r, let 2r denote the property that the girth of a graph is larger than 2r. Then the maximum number of edges of a graph with n vertices, which has property 2r , satisfies 2r ) O(n 1 1 r ) For r = 2, 3 and 5, this bound is tight. ....

....a fixed positive integer r, let 2r denote the property that the girth of a graph is larger than 2r. Then the maximum number of edges of a graph with n vertices, which has property 2r , satisfies 2r ) O(n 1 1 r ) For r = 2, 3 and 5, this bound is tight. 00 . KST54] F96] ER62] [B66], ARS98] For any integers s r 2, the maximum number of edges of a K r,s free graph of n vertices, satisfies ex(n, K r,s ) O(n 2 1 r ) This bound is tight for s (r 1) In case r = 3, we obtain the following slight generalization of Theorem 2. Theorem 3.2. Let G be a graph of n ....

W. G. Brown, On graphs that do not contain a Thomsen graph, Canadian Mathematical Bulletin 9 (1966), 281-285.

....vertices (the rows and columns) are ordered. This is a very important difference but in some special case the restriction on the order is insignificant. An example is the four cycle (complete bipartite graph between two color classes of size 2 each) Classical results in graph theory [KST] ERS] [B] immediately give us the following theorem. Theorem 1.1. f(n; 1 1 1 1 ) Theta(n 3 2 ) We do not know exactly how these two problems are related, but the following facts are known. The Erdos Stone Simonovits theorem ( ESi] ESt] for a survey see Bollob as book [Bo] says that ....

W.G. Brown, On graphs that do not contain a Thomsen graph, Canad. Math. Bull., 9 (1966), 281-285.

....from the fact that Theorem 1 was tight. # Theorems 2 and 3 can be proved similarly. It is enough to notice that splitting a vertex of high degree does not decrease the girth of a graph G and does not create a subgraph isomorphic to K r,s . Instead of Claim C, now we need Claim C 0 . BS74] [B66], Be66] S66] W91] For a fixed positive integer r, let G 2r denote the property that the girth of a graph is larger than 2r. Then the maximum number of edges of a graph with n vertices, which has property G 2r , satisfies ex(n, G 2r ) O(n 1 1 r ) For r = 2, 3 and 5, this bound is ....

....r, let G 2r denote the property that the girth of a graph is larger than 2r. Then the maximum number of edges of a graph with n vertices, which has property G 2r , satisfies ex(n, G 2r ) O(n 1 1 r ) For r = 2, 3 and 5, this bound is tight. Claim C 00 . KST54] F96] ER62] [B66], ARS98] For any integers s # r # 2, the maximum number of edges of a K r,s free graph of n vertices, satisfies ex(n, K r,s ) O(n 2 1 r ) 9 This bound is tight for s (r 1) In case r = 3, we obtain the following slight generalization of Theorem 2. Theorem 3.2. Let G be a ....

W. G. Brown, On graphs that do not contain a Thomsen graph, Canadian Mathematical Bulletin 9 (1966), 281-285.

....from the fact that Theorem 1 was tight. 2 Theorems 2 and 3 can be proved similarly. It is enough to notice that splitting a vertex of high degree does not decrease the girth of a graph G and does not create a subgraph isomorphic to K r,s . Instead of Claim C, now we need Claim C 0 . BS74] [B66], Be66] S66] W91] For a fixed positive integer r, let G 2r denote the property that the girth of a graph is larger than 2r. Then the maximum number of edges of a graph with n vertices, which has property G 2r , satisfies ex(n, G 2r ) O(n 1 1 r ) For r = 2, 3 and 5, this bound is ....

....r, let G 2r denote the property that the girth of a graph is larger than 2r. Then the maximum number of edges of a graph with n vertices, which has property G 2r , satisfies ex(n, G 2r ) O(n 1 1 r ) For r = 2, 3 and 5, this bound is tight. Claim C 00 . KST54] F96] ER62] [B66], ARS98] For any integers s # r # 2, the maximum number of edges of a K r,s free graph of n vertices, satisfies ex(n, K r,s ) O(n 2 1 r ) This bound is tight for s (r 1) 9 In case r = 3, we obtain the following slight generalization of Theorem 2. Theorem 3.2. Let G be ....

W. G. Brown, On graphs that do not contain a Thomsen graph, Canadian Mathematical Bulletin 9 (1966), 281-285.

....B] where the parts A and B do not necessarily have equal sizes, but jAj; jBj = 1 2 n o(n) and one can add an arbitrary graph of girth 5 on A. Obviously, this example carries at most 11 edges in every 6 vertices, and it has 1 4 n 2 Theta(n 3=2 ) edges. For C 4 free graphs see Brown [6] and Erdos, R enyi, S os [21] The density problem ex(n; k; r) was investigated in several more papers, for example in [19] Erdos, Faudree, Jagota and Luczak deal with the case when k = Theta(n) and r = O(n 2 ) A strongly related problem, the restricted multicoloring of the edges of the ....

....ex Z (n; 4; 0) 0, ex Z (n; 4; 1) 1 and ex Z (n; 4; 2) b2n=3c follows immediately. For r = 3 we also get that the weight 1 edges can not contain a C 4 , so that ex Z (n; 4; 3) ex(n; C 4 ) 1 2 n 3=2 O(n 4=3 ) by a result of Erdos, R enyi and S os [21] and independently Brown [6]. Hence ex Z (4; r) 0 for all r 3, but exZ (4; 3) 1=3 which is the only case for k = 4. exZ (4; 4) 1=2 = ex Z (4; 4) and exZ (4; 5) 2=3 = ex Z (4; 5) For k = 5, the cases where exZ (5; r) is bigger than ex Z (5; r) are r = 4; 5; 14. The arguments for r = 4; 5 are similar to those ....

W. G. Brown, On graphs that do not contain a Thomsen graph, Canad. Math. Bull. 9 (1966), 281--285.

.... One of the early problems posed by our beloved Paul Erdos [9] see page ) was that of determining ex(n; C 4 ) Later, Erdos [10] conjectured that ex(n; K 2;2 ) q(q 1) 2 =2 when n = q 2 q 1 and q is a prime power, and this was proved by Furedi for q = 2 k [13] and for q 25 [14] Brown [3] proved that ex(n; K 3;3 ) Theta(n 5=3 ) and Furedi [15] proved that Brown s construction is asymptotically optimal, yielding ex(n; K 3;3 ) 1 2 o(1) n 5=3 . Furedi [16] also generalized the constructions using projective planes to prove that ex(n; K 2;t ) 1 2 p t Gamma 1n ....

W.G. Brown, On graphs that do not contain a Thomsen graph, Canad. Math. Bull. 9(1966), 4 281--285.

.... appears in [16] For a given F , the first problem in studying ex(n; F ) is thus to find the right exponent (if such exists) It is conjectured ( 7,9] that ex(n; K k;k ) Theta(n 2 Gamma1=k ) This was proved for K 2;2 in Erdos R enyi S os [5] and (simultaneously and independently) in Brown [2]. For K 3;3 it appears in Brown [2] For K k;l with k l , results appear in Koll ar R onyai Szab o [15] later improved for k (l Gamma 1) in Alon R onyai Szab o [1] Erdos [6] also proved that when r l 1, there exists a constant c r;l such that ex(n; H 0 (r; l) c r;l n ....

.... the first problem in studying ex(n; F ) is thus to find the right exponent (if such exists) It is conjectured ( 7,9] that ex(n; K k;k ) Theta(n 2 Gamma1=k ) This was proved for K 2;2 in Erdos R enyi S os [5] and (simultaneously and independently) in Brown [2] For K 3;3 it appears in Brown [2]. For K k;l with k l , results appear in Koll ar R onyai Szab o [15] later improved for k (l Gamma 1) in Alon R onyai Szab o [1] Erdos [6] also proved that when r l 1, there exists a constant c r;l such that ex(n; H 0 (r; l) c r;l n 2 Gamma1= r Gammal) His proof is somewhat ....

W. G. Brown, On graphs that do not contain a Thomsen graph, Canad. Math. Bull. 9 (1966), 281--289.

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W.G. Brown. On graphs that do not contain a Thomsen graph. Canad. Math. Bull., 9:281--285, 1966.

No context found.

W. Brown. On graphs that do not contain a Thomsen graph. Canad. Math. Bull., 9:281--285, 1966.

No context found.

W.G. Brown. On graphs that do not contain a Thomsen graph. Canad. Math. Bull., 9:281--285, 1966.

No context found.

Brown, W., On graphs that do not contain a Thomsen graph, Canad. Math. Bull 9 (1966), 281-285.

No context found.

Brown, W., On graphs that do not contain a Thomsen graph, Canad. Math. Bull 9 (1966), 281--285.

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