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The number of Ks,tfree graphs
 Journal of the London Mathematical Society
"... Denote by fn(H) the number of (labeled) Hfree graphs on a fixed vertex set of size n. Erdős conjectured that whenever H contains a cycle, fn(H) = 2 (1+o(1)) ex(n,H) , yet it still open for every bipartite graph, and even the order of magnitude of log 2 fn(H) was known only for C4, C6 and K3,3. We ..."
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Denote by fn(H) the number of (labeled) Hfree graphs on a fixed vertex set of size n. Erdős conjectured that whenever H contains a cycle, fn(H) = 2 (1+o(1)) ex(n,H) , yet it still open for every bipartite graph, and even the order of magnitude of log 2 fn(H) was known only for C4, C6 and K3,3. We show that for all 2 ≤ s ≤ t, fn(Ks,t) = 2 O(n2−1/s) which is asymptotically sharp for all those values of s and t for which the order of magnitude of the Turán number ex(n, Ks,t) is known. Our methods allow us to prove, among other things, that there is a positive constant c, such that almost all K2,tfree graphs of order n have at least 1/12 · ex(n, K2,t) and at most (1 − c) · ex(n, K2,t) edges. Moreover, our results have some interesting applications to the study of some Ramseyand Turántype problems. 1
Graph Powers
 in Contemporary Combinatorics, B. Bollobas
, 2002
"... The investigation of the asymptotic behaviour of various parameters of powers of a fixed graph leads to many fascinating problems, some of which are motivated by questions in information theory, communication complexity, geometry and Ramsey theory. In this survey we discuss these problems and descri ..."
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The investigation of the asymptotic behaviour of various parameters of powers of a fixed graph leads to many fascinating problems, some of which are motivated by questions in information theory, communication complexity, geometry and Ramsey theory. In this survey we discuss these problems and describe the techniques used in their study which combine combinatorial, geometric, probabilistic and linearalgebra tools. 1 Graph Powers There are several known distinct ways to define the powers of a fixed graph. The nth AND power of an undirected graph G = (V, E) is the graph denoted by G ∧n whose vertex set is V n in which distinct vertices (x1... xn) and (x ′ 1... x ′ n) are connected if {xi, x ′ i} ∈ E for all i ∈ {1, 2,..., n} such that xi � = x ′ i. The nth OR power of G is the graph denoted G∨n whose vertex set is V n in which distinct vertices (x1... xn) and (x ′ 1... x ′ n) are connected if distinct xi and x ′ i for some i ∈ {1, 2,..., n}. are connected in G The study of the asymptotic behaviour of various parameters of these powers of a fixed graph G, as well as their behaviour for similarly defined powers of directed and undirected graphs, is motivated by questions in various areas and leads to many intriguing problems. These are discussed in the following sections, in which we focus our attention mainly to the open problems in the area, and only briefly describe the known results and proof techniques. Proven and disproven conjectures are intermingled throughout the paper with open problems. More detailed proofs can be found in the papers listed in the bibliography. 2 Shannon Capacity The independence number α(G) of a graph G is the maximum cardinality of a set of vertices of G no two of which are adjacent.
Some Remarks on Sparsely Connected IsomorphismFree Labeled Graphs
, 2000
"... . Given a set = fH1 ..."
Cleaning a Network with Brushes
"... Following the decontamination metaphor for searching a graph, we introduce a cleaning process, which is related to both the chipfiring game and edge searching. Brushes (instead of chips) are placed on some vertices and, initially, all the edges are dirty. When a vertex is ‘fired’, each dirty incide ..."
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Following the decontamination metaphor for searching a graph, we introduce a cleaning process, which is related to both the chipfiring game and edge searching. Brushes (instead of chips) are placed on some vertices and, initially, all the edges are dirty. When a vertex is ‘fired’, each dirty incident edge is traversed by only one brush, cleaning it, but a brush is not allowed to traverse an already cleaned edge; consequently, a vertex may not need degreemany brushes to fire. The model presented is one where the edges are continually recontaminated, say by algae, so that cleaning is regarded as an ongoing process. Ideally, the final configuration of the brushes, after all the edges have been cleaned, should be a viable starting configuration to clean the graph again. We show that this is possible with the least number of brushes if the vertices are fired sequentially but not if fired in parallel. We also present bounds for the least number of brushes required to clean graphs in general and some specific families of graphs.
The number of Km,mfree graphs
 Combinatorica
"... A graph is called Hfree if it contains no copy of H. Denote by fn(H) the number of (labeled) Hfree graphs on n vertices. Erdős conjectured that fn(H) ≤ 2 (1+o(1)) ex(n,H). This was first shown to be true for cliques; then, Erdős, Frankl, and Rödl proved it for all graphs H with χ(H) ≥ 3. For mos ..."
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A graph is called Hfree if it contains no copy of H. Denote by fn(H) the number of (labeled) Hfree graphs on n vertices. Erdős conjectured that fn(H) ≤ 2 (1+o(1)) ex(n,H). This was first shown to be true for cliques; then, Erdős, Frankl, and Rödl proved it for all graphs H with χ(H) ≥ 3. For most bipartite H, the question is still wide open, and even the correct order of magnitude of log 2 fn(H) is not known. We prove that fn(Km,m) ≤ 2 O(n2−1/m) for every m, extending the result of Kleitman and Winston and answering a question of Erdős. This bound is asymptotically sharp for m ∈ {2, 3}, and possibly for all other values of m, for which the order of ex(n, Km,m) is conjectured to be Θ(n 2−1/m). Our method also yields a bound on the number of Km,mfree graphs with fixed order and size, extending the result of Füredi. Using this bound, we prove a relaxed version of a conjecture due to Haxell, Kohayakawa, and Luczak and show that almost all K3,3free graphs of order n have more than 1/20 · ex(n, K3,3) edges. 1
Almost all C4free graphs have less than (1 − ε) ex(n, C4) edges
, 2009
"... A graph is called Hfree if it contains no copy of H. An old result of Kleitman and Winston [12] states that there are 2 Θ(n3/2) C4free graphs on n vertices. Füredi [8] showed that almost all C4free graphs of order n have at least c · ex(n, C4) edges for some positive constant c> 0. We prove th ..."
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A graph is called Hfree if it contains no copy of H. An old result of Kleitman and Winston [12] states that there are 2 Θ(n3/2) C4free graphs on n vertices. Füredi [8] showed that almost all C4free graphs of order n have at least c · ex(n, C4) edges for some positive constant c> 0. We prove that there is an ε> 0 such that almost all C4free graphs have at most (1 − ε) · ex(n, C4) edges. This resolves a conjecture of Balogh, Bollobás and Simonovits [4] for the 4cycle.
R(3, 4) = 17
"... “Aliens invade the earth and threaten to obliterate it in a year’s time unless human beings can find the Ramsey number for red five and blue five. We could marshal the world’s best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey ..."
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“Aliens invade the earth and threaten to obliterate it in a year’s time unless human beings can find the Ramsey number for red five and blue five. We could marshal the world’s best minds and fastest computers, and within a year we could probably calculate the value. If the aliens demanded the Ramsey number for red six and blue six, however, we would have no choice but to launch a preemptive attack.” Paul Erdős [5] In this paper, we consider the online Ramsey numbers R(k, l) for cliques. Using a high performance computing networks, we ‘calculated ’ that R(3, 4) = 17. We also present an upper bound of R(k, l), study its asymptotic behaviour, and state some open problems. 1 Introduction and
Forbidden subgraphs in connected graphs
, 2004
"... Given a set ξ = {H1,H2, · · ·} of connected non acyclic graphs, a ξfree graph is one which does not contain any member of ξ as copy. Define the excess of a graph as the difference between its number of edges and its number of vertices. Let Wk,ξ be theexponential generating function (EGF for brie ..."
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Given a set ξ = {H1,H2, · · ·} of connected non acyclic graphs, a ξfree graph is one which does not contain any member of ξ as copy. Define the excess of a graph as the difference between its number of edges and its number of vertices. Let Wk,ξ be theexponential generating function (EGF for brief) of connected ξfree graphs of excess equal to k (k ≥ 1). For each fixed ξ, a fundamental differential recurrence satisfied by the EGFs Wk,ξ is derived. We give methods on how to solve this nonlinear recurrence for the first few values of k by means of graph surgery. We also show that for any finite collection ξ of nonacyclic graphs, the EGFs Wk,ξ are always rational functions of the generating function, T, of Cayley’s rooted (nonplanar) labelled trees. From this, we prove that almost all connected graphs with n nodes and n + k edges are ξfree, whenever k = o(n1/3) and ξ  < ∞ by means of Wright’s inequalities and saddle point method. Limiting distributions are derived for sparse connected ξfree components that are present when a random graph on n nodes has approximately n 2
On the StarvsTriangle Size Ramsey Number
, 1999
"... We establish the following bounds on the starvstriangle size ramsey number. n 2 + (0:577 + o(1)) n 3=2 ! r(K 1;n ; K 3 ) ! n 2 + p 2 n 3=2 + n: The upper (constructive) bound disproves a conjecture of Erdos. Introduction A conjecture of Erdos [3], see also e.g. [2, 4], states that ..."
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We establish the following bounds on the starvstriangle size ramsey number. n 2 + (0:577 + o(1)) n 3=2 ! r(K 1;n ; K 3 ) ! n 2 + p 2 n 3=2 + n: The upper (constructive) bound disproves a conjecture of Erdos. Introduction A conjecture of Erdos [3], see also e.g. [2, 4], states that for n 3 any graph with \Gamma 2n+1 2 \Delta \Gamma \Gamma n 2 \Delta \Gamma 1 edges is a union of a bipartite graph and a graph with maximum degree less than n. This value arises from the consideration of P n+1;n which doesn't admit the above representation. (P m;n = Km+E n has m+ n vertices of which m vertices are connected to every other vertex.) In the arrowing notation the latter statement reads "P n+1;n ! (K 1;n ; C odd )": for any bluered colouring of the edgeset of P n+1;n we necessarily have either a blue star K 1;n or a red cycle of odd length. (By C odd we denote the family of odd cycles.) Thus the conjecture states that r(K 1;n ; C odd ) = e(P n+1;n ) and, if true, g...