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• W2052850567 abstract "A graph is Hamiltonian if it contains a cycle which passes through every vertex of the graph exactly once. A classical theorem of Dirac from 1952 asserts that every graph on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> vertices with minimum degree at least <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n slash 2> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>n/2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is Hamiltonian. We refer to such graphs as Dirac graphs. In this paper we extend Dirac’s theorem in two directions and show that Dirac graphs are robustly Hamiltonian in a very strong sense. First, we consider a random subgraph of a Dirac graph obtained by taking each edge independently with probability <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding=application/x-tex>p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and prove that there exists a constant <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C> <mml:semantics> <mml:mi>C</mml:mi> <mml:annotation encoding=application/x-tex>C</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=p greater-than-or-equal-to upper C log n slash n> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mi>C</mml:mi> <mml:mi>log</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>n</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>p ge C log n / n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then a.a.s. the resulting random subgraph is still Hamiltonian. Second, we prove that if a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis 1 colon b right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>:</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(1:b)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Maker-Breaker game is played on a Dirac graph, then Maker can construct a Hamiltonian subgraph as long as the bias <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=b> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding=application/x-tex>b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is at most <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c n slash log n> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mi>n</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>log</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>cn /log n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some absolute constant <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c greater-than 0> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>c > 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Both of these results are tight up to a constant factor, and are proved under one general framework." @default.
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• W2052850567 date "2014-02-10" @default.
• W2052850567 modified "2023-09-30" @default.
• W2052850567 title "Robust Hamiltonicity of Dirac graphs" @default.
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