Results 1  10
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27
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 24 (5 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.
Generalizations of a RamseyTheoretic Result of Chvátal
, 1983
"... 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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Cited by 16 (1 self)
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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.
Ramsey Numbers for Cycles in Graphs
, 1973
"... ... integer m such that, for any partition (E1, E2) of the edges of K,,, , either G 1 is a subgraph of the graph induced by E1, or G2 is a subgraph of the graph induced by E2. We show that R(C „ , Cn) = 2n 1 if n is odd, R(Cs, C2,_,) = 2n I if n> r(2r 1), R(C„,C2,)=n+r1 if n>4r 2r+2, R( ..."
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Cited by 6 (0 self)
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... integer m such that, for any partition (E1, E2) of the edges of K,,, , either G 1 is a subgraph of the graph induced by E1, or G2 is a subgraph of the graph induced by E2. We show that R(C „ , Cn) = 2n 1 if n is odd, R(Cs, C2,_,) = 2n I if n> r(2r 1), R(C„,C2,)=n+r1 if n>4r 2r+2, R(Cn, K,) v nr 2 for all r, n, R(Cn,K,.)=(r1)(n1) +1 if n>r22, R(Cn, K,+1) = t(n 1) + r for large n.
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 6 (3 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
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 4 (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.
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 3 (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...
Recent developments in graph Ramsey theory
"... Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any twocolouring of the edges of KN contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been ..."
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Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any twocolouring of the edges of KN contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. In this survey, we will describe some of this progress.
The Ramsey numbers of large cycles versus small wheels,Integer: The Electronic
 J. of Combinatorial Number Theory
, 2004
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Ramsey Numbers Involving Cycles
"... We gather and review general results and data on Ramsey numbers involving cycles. This survey is based on the author’s 2009 revision #12 of the Dynamic Survey DS1, "Small Ramsey Numbers", at the ..."
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We gather and review general results and data on Ramsey numbers involving cycles. This survey is based on the author’s 2009 revision #12 of the Dynamic Survey DS1, "Small Ramsey Numbers", at the