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The regularity method for graphs and digraphs
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abstract
This MSci thesis surveys results in extremal graph theory, in particular relating to Hamilton cycles. Szem\'eredi's Regularity Lemma plays a central role. We also investigate the robust outexpansion property for digraphs. Kelly showed that every sufficiently large oriented graph on $n$ vertices with minimum in- and outdegree at least $3n/8 +o(n)$ contains any orientation of a Hamilton cycle. We use Kelly's arguments to extend his result to any robustly expanding digraph of linear degree.
Forward citations
Cited by 3 Pith papers
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A directed Andr\'asfai-Erd\H{o}s-S\'os theorem and chromatic profiles of oriented cycles
For every r≥3, the exact chromatic profile of the transitive tournament T_r is (3r-7)/(3r-4); directed odd cycles have 2-color profile 1/2, and the three non-directed pentagon orientations have 2-color profile 1/3.
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Arbitrary orientations of Hamilton cycles in directed graphs of large minimum degree
A minimum degree of (1+o(1))n in an n-vertex digraph forces every orientation of a Hamilton cycle, except the directed cycle when the graph is not strongly connected.
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On the supersaturation of oriented Tur\'an problems
Oriented graphs that exceed the oriented Turán density contain a positive fraction of the possible copies of the forbidden oriented subgraph, with explicit bounds for transitive tournaments and antidirected complete b...
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