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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
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
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.
Almost all C4free graphs have fewer than (1− ε) ex(n,C4) edges
"... A graph is called Hfree if it contains no copy of H. Let ex(n,H) denote the Turán number for H, i.e., the maximum number of edges that an nvertex Hfree graph may have. An old result of Kleitman and Winston states that there are 2O(ex(n,C4)) C4free graphs on n vertices. Füredi showed that almos ..."
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A graph is called Hfree if it contains no copy of H. Let ex(n,H) denote the Turán number for H, i.e., the maximum number of edges that an nvertex Hfree graph may have. An old result of Kleitman and Winston states that there are 2O(ex(n,C4)) C4free graphs on n vertices. Füredi showed that almost all C4free graphs of order n have at least c ex(n,C4) edges for some positive constant c. We prove that there is a positive constant ε such that almost all C4free graphs have at most (1 − ε) ex(n,C4) edges. This resolves a conjecture of Balogh, Bollobás, and Simonovits for the 4cycle.
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.
Some Remarks on Sparsely Connected IsomorphismFree Labeled Graphs
, 2000
"... . Given a set = fH1 ..."
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
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"... Abstract. In this note, 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. “Aliens invade the earth and ..."
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Abstract. In this note, 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. “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 Erdös [5] 1. Introduction and
Choosability of Graphs with Bounded Order: Ohba’s Conjecture and Beyond
, 2013
"... c©Jonathan A. Noel, 2013 The choice number of a graph G, denoted ch(G), is the minimum integer k such that for any assignment of lists of size k to the vertices of G, there is a proper colouring of G such that every vertex is mapped to a colour in its list. For general graphs, the choice number is ..."
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c©Jonathan A. Noel, 2013 The choice number of a graph G, denoted ch(G), is the minimum integer k such that for any assignment of lists of size k to the vertices of G, there is a proper colouring of G such that every vertex is mapped to a colour in its list. For general graphs, the choice number is not bounded above by a function of the chromatic number. In this thesis, we prove a conjecture of Ohba which asserts that ch(G) = χ(G) whenever V (G)  ≤ 2χ(G) + 1. We also prove a strengthening of Ohba’s Conjecture which is best possible for graphs on at most 3χ(G) vertices, and pose several conjectures related to our work. ii Abrégé Le nombre de choix d’un graphe G, note ́ ch(G), est le plus petit entier k tel que pour toute affectation de listes de taille k au sommets de G, il y a une coloration de G tel que chaque sommet de G est colore ́ par une couleur de sa liste. En général, le nombre de choix n’est pas borne ́ supérieurement par une fonction du nombre chromatique. Dans cette thèse, nous démontrons une conjecture de Ohba qui affirme que ch(G) = χ(G) dès que V (G)  ≤ 2χ(G) + 1. Nous démontrons aussi une version plus forte de la conjecture de Ohba qui est optimale pour les graphes ayant au plus 3χ(G) sommets, et énonçons plusieurs conjectures par rapport a ̀ nos travaux. iii Declaration This thesis contains no material which has been accepted in whole, or in part, for any other degree or diploma. Chapters 4 and 6 of this thesis contain new contributions to knowledge. The results of these chapters have been, or will be, submitted for publication in peerreviewed journals. The result of Chapter 4 is based on joint work with Bruce A. Reed and Hehui Wu. The result of Chapter 6 is based on joint work with Douglas B. West, Hehui Wu, and Xuding Zhu. iv