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Characterisations of Nowhere Dense Graphs
, 2013
"... Nowhere dense classes of graphs were introduced by Nešetřil and Ossona de Mendez as a model for “sparsity” in graphs. It turns out that nowhere dense classes of graphs can be characterised in many different ways and have been shown to be equivalent to other concepts studied in areas such as (finite) ..."
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Nowhere dense classes of graphs were introduced by Nešetřil and Ossona de Mendez as a model for “sparsity” in graphs. It turns out that nowhere dense classes of graphs can be characterised in many different ways and have been shown to be equivalent to other concepts studied in areas such as (finite
The Ramsey number of dense graphs
"... The Ramsey number r(H) of a graph H is the smallest number n such that, in any twocolouring of the edges of Kn, there is a monochromatic copy of H. We study the Ramsey number of graphs H with t vertices and density ρ, proving that r(H) ≤ 2 c √ ρ log(2/ρ)t. We also investigate some related problems ..."
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Cited by 4 (4 self)
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The Ramsey number r(H) of a graph H is the smallest number n such that, in any twocolouring of the edges of Kn, there is a monochromatic copy of H. We study the Ramsey number of graphs H with t vertices and density ρ, proving that r(H) ≤ 2 c √ ρ log(2/ρ)t. We also investigate some related
Small minors in dense graphs
 European J. Combin
"... Abstract. A fundamental result in structural graph theory states that every graph with large average degree contains a large complete graph as a minor. We prove this result with the extra property that the minor is small with respect to the order of the whole graph. More precisely, we describe funct ..."
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Abstract. A fundamental result in structural graph theory states that every graph with large average degree contains a large complete graph as a minor. We prove this result with the extra property that the minor is small with respect to the order of the whole graph. More precisely, we describe
Pebbling in Dense Graphs
, 2008
"... A configuration of pebbles on the vertices of a graph is solvable if one can place a pebble on any given root vertex via a sequence of pebbling steps. The pebbling number of a graph G is the minimum number π(G) so that every configuration of π(G) pebbles is solvable. A graph is Class 0 if its pebbli ..."
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Cited by 4 (3 self)
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A configuration of pebbles on the vertices of a graph is solvable if one can place a pebble on any given root vertex via a sequence of pebbling steps. The pebbling number of a graph G is the minimum number π(G) so that every configuration of π(G) pebbles is solvable. A graph is Class 0 if its
Spanning Trees in Dense Graphs
, 2001
"... In this paper we prove the following almost optimal theorem. For any δ>0, there exist constants c and n0 such that, if n � n0, T is a tree of order n and maximum degree at most cn / log n, and G is a graph of order n and minimum degree at least (1/2+δ)n, then T is a subgraph of G. ..."
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Cited by 7 (0 self)
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In this paper we prove the following almost optimal theorem. For any δ>0, there exist constants c and n0 such that, if n � n0, T is a tree of order n and maximum degree at most cn / log n, and G is a graph of order n and minimum degree at least (1/2+δ)n, then T is a subgraph of G.
Packing and covering of dense graphs
 Journal of Combinatorial Designs
, 1998
"... Let d be a positive integer. A graph G is called ddivisible if d divides the degree of each vertex of G. G is called nowhere ddivisible if no degree of a vertex of G is divisible by d. For a graph H, gcd(H) denotes the greatest common divisor of the degrees of the vertices of H. The Hpacking numb ..."
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Cited by 5 (2 self)
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Let d be a positive integer. A graph G is called ddivisible if d divides the degree of each vertex of G. G is called nowhere ddivisible if no degree of a vertex of G is divisible by d. For a graph H, gcd(H) denotes the greatest common divisor of the degrees of the vertices of H. The H
Cycle Factors in Dense Graphs
 DISCRETE MATHEMATICS 197/198 (1999), 309323, 16TH BRITISH COMBINATORIAL CONFERENCE
, 1999
"... It is proven that a graph with n vertices and minimum degree at least h+2 2h n contains n h \Gamma O(h 2 ) vertex disjoint cycles of size h, and that a graph with n ? N (h) vertices and minimum degree at least h+3 2h n contains n h vertex disjoint hcycles, provided h divides n. Bounds on ..."
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Cited by 2 (1 self)
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It is proven that a graph with n vertices and minimum degree at least h+2 2h n contains n h \Gamma O(h 2 ) vertex disjoint cycles of size h, and that a graph with n ? N (h) vertices and minimum degree at least h+3 2h n contains n h vertex disjoint hcycles, provided h divides n. Bounds
Irregularity Strength of Dense Graphs
, 2008
"... Let G be a simple graph of order n with no isolated vertices and no isolated edges. For a positive integer w, an assignment f on G is a function f: E(G) → {1, 2,..., w}. For a vertex v, f(v) is defined as the sum f(e) over all edges e of G incident with v. f is called irregular, if all f(v) are dist ..."
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Cited by 3 (0 self)
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Let G be a simple graph of order n with no isolated vertices and no isolated edges. For a positive integer w, an assignment f on G is a function f: E(G) → {1, 2,..., w}. For a vertex v, f(v) is defined as the sum f(e) over all edges e of G incident with v. f is called irregular, if all f
On perfect packings in dense graphs
, 2011
"... We say that a graph G has a perfect Hpacking if there exists a set of vertexdisjoint copies of H which cover all the vertices in G. We consider various problems concerning perfect Hpackings: Given n, r, D ∈ N, we characterise the edge density threshold that ensures a perfect Krpacking in any gra ..."
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We say that a graph G has a perfect Hpacking if there exists a set of vertexdisjoint copies of H which cover all the vertices in G. We consider various problems concerning perfect Hpackings: Given n, r, D ∈ N, we characterise the edge density threshold that ensures a perfect Krpacking in any
Results 11  20
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168,018