P. Erdos, A. Renyi, and V.T. Sos. On a problem of graph theory. Studia Sci. Math. Hungar., 1:215--235, 1966.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....closed or n e.c. if for every n element subset S of the vertices, and for every subset T of S, there is a vertex x S which is directed toward every vertex in T and directed away from every vertex in S T. Note that T may be empty. Adjacency properties of tournaments were studied in [3, 9, 15, 18, 22]. Much of the research on such properties is motivated by the fact that while almost all tournaments (with arcs chosen independently and with probability p, where 0 p 1 is a fixed real number) are n e.c. for any fixed positive integer n (see [15] few explicit examples of such tournaments are ....

....of tournaments were studied in [3, 9, 15, 18, 22] Much of the research on such properties is motivated by the fact that while almost all tournaments (with arcs chosen independently and with probability p, where 0 p 1 is a fixed real number) are n e.c. for any fixed positive integer n (see [15]) few explicit examples of such tournaments are known. Adjacency properties of graphs were studied by numerous authors; see [10] for a survey. A graph is called n existentially closed or n e.c. if it satisfies the following adjacency property: for every n element subset 1991 Mathematics Subject ....

P. Erdos, On a problem in graph theory, Math. Gaz. 47 (1963), 220-223.

....in V ( T ) and every orientation vector of length k 0 , the sequence S has at least 2 k Gammak 0 successors in T . This is obvious since any orientation vector of length k 0 can be extended to 2 k Gammak 0 disctinct orientation vectors of length k. Schutte and Erdos (see [7] or [12] pp. 28 31) considered a similar property of tournaments, the so called property S(k) by only requiring the existence of a successor for the orientation vector = 1; 1; 1) Erdos proved in [7] that for every k, there exist finite tournaments satisfying the property S(k) In ....

....to 2 k Gammak 0 disctinct orientation vectors of length k. Schutte and Erdos (see [7] or [12] pp. 28 31) considered a similar property of tournaments, the so called property S(k) by only requiring the existence of a successor for the orientation vector = 1; 1; 1) Erdos proved in [7] that for every k, there exist finite tournaments satisfying the property S(k) In fact, by slightly modifying Erdos s proof, we get the following Theorem 4 For every k 0, there exists a finite tournament T k satisfying the property P (k) Proof. Let n be such that 2 k i n k j (1 Gamma ....

P. Erdos, On a problem in graph theory, Math. Gaz. 47 (1963), 220--223.

....graphs of order n, diameter 2 and maximal degree bpnc (0 p 1) and put F 2 (n; bpnc) min G2H2 (n;bpnc) e(G) where e(G) is the size of G. Denote further by f(p) lim n 1 F 2 (n; bpnc) The function f(p) was introduced in [1] and in [5] the existence of the limit (conjectured in [2]) was proved for all values of p except of a sequence tending to 0. It is also showed in [5] that for a given p, f(p) can be determined using linear programming. However, this procedure is too slow to enable us to solve the problem even for relatively large values of p. In [2] the values of f(p) ....

....(conjectured in [2] was proved for all values of p except of a sequence tending to 0. It is also showed in [5] that for a given p, f(p) can be determined using linear programming. However, this procedure is too slow to enable us to solve the problem even for relatively large values of p. In [2] the values of f(p) for p 1=2 were determined. Further, in [4] it was shown that if a projective plane of order t exists, then f(p) t 1 for (t 1) t 2 t 1) p 1=t, hence putting t = 2 we get f(p) 3 if 3=7 p 1=2. In [6] f(p) is determined for 5=12 p 3=7. Thus for all p ....

Erdos P., R'enyi A. and T. S'os V., On a problem in graph theory, Studia Sci. Math. Hungar. 1 (1966), 215--235.

....the covering number of H . A hypergraph is called critical if omitting any one of its edges reduces its covering number. Several problems in Extremal Set Theory deal with the estimation of the size of critical systems of given types (see [2] 3] for various examples) Erdos, Hajnal and Moon [1] proved that any critical graph has at most Gamma 1 2 Delta edges. As a generalization of this result J.P. Roudneff [2] as well as R. Aharoni and R.Ziv [3] conjectured that the same estimate holds for any critical linear hypergraph. In this note we prove this conjecture for 5. We ....

P. Erdos, A. Hajnal and J. Moon, A problem in graph theory, Amer. Math. Monthly 71 (1964), 1107-1110.

....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 complete graph Kn , was ....

....2 in the positive weight case. 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 = ....

P. Erdos, A. R'enyi, V. T. S'os, On a problem of graph theory, Studia Sci. Math. Hungar. 1 (1966), 215--235.

....setting. Our lower bound depends on a bound for another problem. We say that a vertex of a tournament dominates all its successors, and a tournament is k dominated (historically, satisfies property P k ) if for every k set U of vertices there is a vertex outside U that dominates U . Erdos [2] proved that when n is sufficiently large, some n vertex tournament is k dominated, because then the expected number of undominated k sets in the random tournament is less than 1. The expected number of undominated k sets is Gamma n k Delta (1 Gamma 2 Gammak ) n Gammak . Since Gamma ....

P. Erdos, On a problem in graph theory, Math. Gaz. 47(1963), 220--223.

....dominator of y i . Definition 1.1 A time stamp system of order k is a directed graph, in which any ordered sequence having less than k elements has a dominator. In such a graph, any vertex belongs to an ordered sequence of cardinality k. A related notion was considered by many authors after Erdos [7, 8, 13], namely, that of a tournament (i.e. a directed graph in which for any pair of vertices fx; yg one is the dominator of the other) satisfying the so called property S(p) For such a tournament, any subset of cardinality p has a dominator. Hence, any tournament with property S(p) is a time stamp ....

P. Erdos, On a problem in graph theory, Math. Gaz. 47 (1963); 220 \Gamma 223.

....j i, x fi = y ff if fi ff, x fi = 1 Gamma y ff otherwise, if j i, x fi = 1 Gamma y ff if fi ff, x fi = y ff otherwise. 2 5 Rotational tournaments The notion of time stamp system is related to a property of tournaments, stronger than (E k ) initially considered by Schutte and Erdos [3, 8] : a tournament T satisfies the property (S k ) is every set having at most k Gamma 1 vertices in T has a successor (in case of time stamp systems, only ordered sequences are considered) Szekeres and Szekeres [10] have proved that such a tournament must have at least 2 k Gamma2 (k 1) Gamma ....

P. Erdos, On a problem in graph theory, Math. Gaz. 47 (1963), 220-223.

.... ) this was observed by Erdos [unpublished] and 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 ....

P. Erdos, A. R'enyi, and V. T. S'os, On a problem of graph theory, Studia Sci. Math. Hungar. 1 (1966), 215--235. Also see [8], 201--221.

....in a tournament can be found in njO(log n) time. Proof: In view of Fact 2.5, a minimum dominating set can be found by enumerating all subsets of V of cardinality not greater than dlog 2 ne. There are P i=1 jdlog 2 ne Gamma n i Delta such subsets and this establishes the proof. P. Erdos [E] used the probabilistic method to prove the following fact: Fact 2.5. For every ffl 0 there is a number K such that for every k K there exists a tournament T k with no more than 2jkkj2 log(2 ffl) vertices such that (T k ) k. Proof: Consider a random tournament T with n vertices, that is, ....

P. Erdos, "On a problem in graph theory", Math. Gaz. 47 (1963) 220-223.

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P. Erdos, A. Renyi, and V.T. Sos. On a problem of graph theory. Studia Sci. Math. Hungar., 1:215--235, 1966.

No context found.

P. Erdos, "On a problem in graph theory", Math. Gaz. 47(19fi 220-223.

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P. Erdos, A. Renyi, and V. Sos. On a problem of graph theory. Studia Sci. Math. Hungar., 1:215--235, 1966.

No context found.

P. Erdos, A. R'enyi, and V.T. S'os. On a problem of graph theory. Studia Sci. Math. Hungar., 1:215--235, 1966.

No context found.

P. Erdos, A. Renyi, and V. Sos. On a problem of graph theory. Studia Sci. Math. Hungar., 1:215--235, 1966.

No context found.

P. Erdos, On a problem in graph theory, Math. Gaz. 47 (1963), 220-223.

No context found.

Erdos, P., Renyi, A., T. Sos, V., On a problem of graph theory, Studia Sci.Math. Hungar. 1 (1966), 51--57.

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