Results 1  10
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15
Graph Ramsey theory and the polynomial hierarchy (Extended Abstract)
 ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING (ATLANTA, GA, 1999), ACM
, 1999
"... In the Ramsey theory of graphs F + (G, H) means that for every way of coloring the edges of F red and blue F will contain either a red G or a blue H. The problem ARROWING of deciding whether F + (G, H) lies in II; = coNPNP and it was shown to be coNP hard by Burr [5]. We prove that ARROWING is actu ..."
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Cited by 19 (4 self)
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In the Ramsey theory of graphs F + (G, H) means that for every way of coloring the edges of F red and blue F will contain either a red G or a blue H. The problem ARROWING of deciding whether F + (G, H) lies in II; = coNPNP and it was shown to be coNP hard by Burr [5]. We prove that ARROWING is actually II;complete, simultaneously settling a conjecture of Burr and providing a natural example of a problem complete for a higher level of the polynomial hierarchy. We also show that STRONG ARROWING, the version for induced subgraphs, is rI;complete.
LoeblKomlós–Sós Conjecture: dense case
"... We prove a version of the LoeblKomlós–Sós Conjecture for dense graphs.For any q> 0 there exists a number n0 2 N such that for any n> n0 and k> qn the following holds: if G be a graph of order n with at least n ..."
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Cited by 3 (2 self)
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We prove a version of the LoeblKomlós–Sós Conjecture for dense graphs.For any q> 0 there exists a number n0 2 N such that for any n> n0 and k> qn the following holds: if G be a graph of order n with at least n
The Ramsey Numbers of Large Cycles versus Small Wheels
"... For two given graphs G and H, the Ramsey number R(G, H) is the smallest positive integer N such that for every graph F of order N the following holds: either F contains G as a subgraph or the complement of F contains H as a subgraph. In this paper, we determine the Ramsey number R(C,W,,) for m = ..."
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Cited by 2 (0 self)
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For two given graphs G and H, the Ramsey number R(G, H) is the smallest positive integer N such that for every graph F of order N the following holds: either F contains G as a subgraph or the complement of F contains H as a subgraph. In this paper, we determine the Ramsey number R(C,W,,) for m = 4 and m = 5. We show that R(C,W4) = 2n  1 and R(C, Wx) = 3n  2 for n _> 5. For larger wheels it remains an open problem to determine R(C, W,,).
A Conjecture of Erdös the Ramsey Number r(W6)
 J. Combinatorial Math. and Combinatorial Computing
, 1996
"... It was conjectured by Paul Erdos that if G is a graph with chromatic number at least k; then the diagonal Ramsey number r(G) r(K k ). That is, the complete graph K k has the smallest diagonal Ramsey number among the graphs of chromatic number k. This conjecture is shown to be false for k = 4 by ve ..."
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Cited by 2 (0 self)
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It was conjectured by Paul Erdos that if G is a graph with chromatic number at least k; then the diagonal Ramsey number r(G) r(K k ). That is, the complete graph K k has the smallest diagonal Ramsey number among the graphs of chromatic number k. This conjecture is shown to be false for k = 4 by verifying that r(W 6 ) = 17; where W 6 is the wheel with 6 vertices, since it is well known that r(K 4 ) = 18. Computational techniques are used to determine r(W 6 ) as well as the Ramsey numbers for other pairs of small order wheels. 1 Introduction The following well known conjecture is due to Paul Erdos. CONJECTURE 1 If G is a graph with chromatic number Ø(G) k; then the Ramsey number r(G) r(K k ): The strong form of the Erdos conjecture is that if Ø(G) k; and G does not contain a copy of K k ; then r(G) ? r(K k ). For k = 3 it is trivial to verify this stronger conjecture. If G 6' K 3 and Ø(G) 3; then G has at least 4 vertices. Thus r(G) ? 6 = r(K 3 ); since neither the graph K 3 [K...
Proof of the (n/2 − n/2 − n/2) conjecture for large n
"... A conjecture of Loebl, known as the (n/2 − n/2 − n/2) Conjecture, states that if G is an nvertex graph in which at least n/2 of the vertices have degree at least n/2, then G contains all trees with at most n/2 edges as subgraphs. Applying the Regularity Lemma, Ajtai, Komlós and Szemerédi proved an ..."
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Cited by 2 (0 self)
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A conjecture of Loebl, known as the (n/2 − n/2 − n/2) Conjecture, states that if G is an nvertex graph in which at least n/2 of the vertices have degree at least n/2, then G contains all trees with at most n/2 edges as subgraphs. Applying the Regularity Lemma, Ajtai, Komlós and Szemerédi proved an approximate version of this conjecture. We prove it exactly for sufficiently large n and show that this is essentially best possible. Our result gives a tight bound for the Ramsey number of trees, and thus partially answers a conjecture of Burr and Erdős. Along with our proof, we also give a stability theorem, which describe the structure of nvertex graphs that have at least (1 − ε)n/2 vertices of degree at least n/2 but does not contain certain tree with n/2 edges. 1.
The Ramsey Numbers of Large StarLike Trees versus Large Odd Wheels
, 2002
"... For two given graphs G and H, the Ramsey number R(G, H) is the smallest positive integer N such that for every graph F of order N the following holds: either F contains G as a subgraph or the complement of F contains H as a subgraph. In this paper, we shall study the Ramsey number R(T, W,) for a ..."
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Cited by 1 (1 self)
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For two given graphs G and H, the Ramsey number R(G, H) is the smallest positive integer N such that for every graph F of order N the following holds: either F contains G as a subgraph or the complement of F contains H as a subgraph. In this paper, we shall study the Ramsey number R(T, W,) for a starlike tree T with n vertices and a wheel W, with m + 1 vertices and m odd. We show that the Ramsey number R(S,W,)  3n  2 for n _ 2m  4, m _ 5 and m odd, where S denotes the star on n vertices. We conjecture that the Ramsey number is the same for general trees on n vertices, and support this conjecture by proving it for a number of starlike trees.
Size of Monochromatic Double Stars in Edge Colorings
"... We show that in every rcoloring of the edges of Kn there is a monochromatic double star with at least n(r+1)+r−1 r 2 vertices. This result is sharp in asymptotic for r = 2and for r ≥ 3 improves a bound of Mubayi for the largest monochromatic subgraph of diameter at most three. When rcolorings are ..."
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We show that in every rcoloring of the edges of Kn there is a monochromatic double star with at least n(r+1)+r−1 r 2 vertices. This result is sharp in asymptotic for r = 2and for r ≥ 3 improves a bound of Mubayi for the largest monochromatic subgraph of diameter at most three. When rcolorings are replaced by local rcolorings, our bound is n(r+1)+r−1 r 2. +1
Generalizations of a RamseyTheoretic Result of Chv6tal
"... Chvátal has shown that if T is a tree on n points then r(Kk, T) _ (k 1) (n 1) + 1, where r is the (generalized) Ramsey number. It is shown that the same result holds when T is replaced by many other graphs. Such a T is called kgood. The results proved all support the conjecture that any large gr ..."
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Chvátal has shown that if T is a tree on n points then r(Kk, T) _ (k 1) (n 1) + 1, where r is the (generalized) Ramsey number. It is shown that the same result holds when T is replaced by many other graphs. Such a T is called kgood. The results proved all support the conjecture that any large graph that is sufficiently sparse, in the appropriate sense, is kgood. 1
RAMSEY THEOREMS FOR MULTIPLE COPIES OF GRAPHS BY
, 1975
"... to be the least number p such that if the edges of the complete graph colored red and blue (say), either the red graph contains Kp are G as a subgraph or the blue graph contains H. Let mG denote the union of m disjoint copies of G. following result is proved: Let G and H have k and I points respecti ..."
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to be the least number p such that if the edges of the complete graph colored red and blue (say), either the red graph contains Kp are G as a subgraph or the blue graph contains H. Let mG denote the union of m disjoint copies of G. following result is proved: Let G and H have k and I points respectively and have point independence numbers of i and j respectively. Then N 1 5 r(mG, nH) < N + C, where N = km + In min(mi, ml) and where C is an effectively computable function of G and H. The method used permits exact evaluation of r(mG, nH) for various choices of G and H, especially when m = n or G = H. In particular, r(mK3, nK 3) = 3m + 2n when m _> n, m 3 2. 1. Introduction. Let G and H be graphs without isolated points. Following Chvátal and Harary [11, define the Ramsey number r(G, H) to be the least integer n such that if the edges of Kn (the complete graph on n points) are twocolored,
On Some Multicolor Ramsey Numbers Involving K3 + e and K4 − e
"... Abstract: The Ramsey number R(G1, G2, G3) is the smallest positive integer n such that for all 3colorings of the edges of Kn there is a monochromatic G1 in the first color, G2 in the second color, or G3 in the third color. We study the bounds on various 3color Ramsey numbers R(G1, G2, G3), where G ..."
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Abstract: The Ramsey number R(G1, G2, G3) is the smallest positive integer n such that for all 3colorings of the edges of Kn there is a monochromatic G1 in the first color, G2 in the second color, or G3 in the third color. We study the bounds on various 3color Ramsey numbers R(G1, G2, G3), where Gi ∈ {K3, K3 + e, K4 − e, K4}. The minimal and maximal combinations of Gi’s correspond to the classical Ramsey numbers R3(K3) and R3(K4), respectively, where R3(G) = R(G, G, G). Here, we focus on the much less studied combinations between these two cases. Through computational and theoretical means we establish that R(K3, K3, K4 − e) = 17, and by construction we raise the lower bounds on R(K3, K4 − e, K4 − e) and R(K4, K4 − e, K4 − e). For some G and H it was known that R(K3, G, H) = R(K3 + e, G, H); we prove this is true for several more cases including R(K3, K3, K4 −e) = R(K3 +e, K3 +e, K4 −e). Ramsey numbers generalize to more colors, such as in the famous 4color case of R4(K3), where monochromatic triangles are avoided. It is known that 51 ≤ R4(K3) ≤ 62. We prove a surprising theorem stating that if R4(K3) = 51 then R4(K3 + e) = 52, otherwise R4(K3 + e) = R4(K3). 1