A counterexample to a conjecture of grant

Cai, Mao-Cheng

doi:10.1016/0012-365x(83)90010-9

Cited by 9 publications

(9 citation statements)

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“…This means that we can find an increasing function f for which all of the conditions in the proof are satisfied whenever a ≤ f (b). It is equivalent to setting a := min{f 1…”

Section: Notationmentioning

confidence: 99%

Arbitrary Orientations of Hamilton Cycles in Digraphs

DeBiasio¹,

Kühn²,

Molla³

et al. 2015

SIAM J. Discrete Math.

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Abstract. Let n be sufficiently large and suppose that G is a digraph on n vertices where every vertex has in-and outdegree at least n/2. We show that G contains every orientation of a Hamilton cycle except, possibly, the antidirected one. The antidirected case was settled by DeBiasio and Molla, where the threshold is n/2 + 1. Our result is best possible and improves on an approximate result by Häggkvist and Thomason.

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“…This means that we can find an increasing function f for which all of the conditions in the proof are satisfied whenever a ≤ f (b). It is equivalent to setting a := min{f 1…”

Section: Notationmentioning

confidence: 99%

Arbitrary Orientations of Hamilton Cycles in Digraphs

DeBiasio¹,

Kühn²,

Molla³

et al. 2015

SIAM J. Discrete Math.

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“…If there exists x 1 ∈ X 1 and x 2 ∈ X 2 such that x i x 3−i is not an edge for some i ∈ [2], then because of the semi-degree condition and the fact that X 1 and X 2 are independent sets, it must be that x i y i and x 3−i y 3−i are edges, giving us two independent connecting edges. If there exists x i ∈ X i such that y i x i is not an edge, then, by the semi-degree condition and the fact that X i is an independent set, y 3−i x i is an edge.…”

Section: Finding the Adhcmentioning

confidence: 99%

“…However, in 1983, Cai [2] gave a counterexample to Grant's conjecture (see Figure 1b). Example 1.2 (Cai 1983). For all n, there exists a directed graph D on 2n vertices with δ 0 (D) = n such that D does not contain an ADHC.…”

Section: Introductionmentioning

confidence: 96%

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Semi-degree threshold for anti-directed Hamiltonian cycles

DeBiasio

Molla

2013

Electronic Notes in Discrete Mathematics

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In 1960 Ghouila-Houri extended Dirac's theorem to directed graphs by proving that if D is a directed graph on n vertices with minimum out-degree and in-degree at least n/2, then D contains a directed Hamiltonian cycle. For directed graphs one may ask for other orientations of a Hamiltonian cycle and in 1980 Grant initiated the problem of determining minimum degree conditions for a directed graph D to contain an anti-directed Hamiltonian cycle (an orientation in which consecutive edges alternate direction). We prove that for sufficiently large even n, if D is a directed graph on n vertices with minimum out-degree and in-degree at least n 2 + 1, then D contains an anti-directed Hamiltonian cycle. In fact, we prove the stronger result that n 2 is sufficient unless D is one of two counterexamples. This result is sharp.

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“…In his paper Grant [7] conjectured that the theorem above can be strengthened to assert that if D is a directed graph with even order n and if δ 0 (D) 1 2 n then D contains an anti-directed Hamilton cycle. Mao-cheng Cai [10] gave a counter-example to this conjecture. However, the following theorem by Häggkvist and Thomason [8] proves that Grant's conjecture is asymptotically true.…”

Section: Theorem 3 [7]mentioning

confidence: 99%

Vertex-Oriented Hamilton Cycles in Directed Graphs

Plantholt¹,

Tipnis²

2009

Electron. J. Combin.

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Let $D$ be a directed graph of order $n$. An anti-directed Hamilton cycle $H$ in $D$ is a Hamilton cycle in the graph underlying $D$ such that no pair of consecutive arcs in $H$ form a directed path in $D$. We prove that if $D$ is a directed graph with even order $n$ and if the indegree and the outdegree of each vertex of $D$ is at least ${2\over 3}n$ then $D$ contains an anti-directed Hamilton cycle. This improves a bound of Grant. Let $V(D) = P \cup Q$ be a partition of $V(D)$. A $(P,Q)$ vertex-oriented Hamilton cycle in $D$ is a Hamilton cycle $H$ in the graph underlying $D$ such that for each $v \in P$, consecutive arcs of $H$ incident on $v$ do not form a directed path in $D$, and, for each $v \in Q$, consecutive arcs of $H$ incident on $v$ form a directed path in $D$. We give sufficient conditions for the existence of a $(P,Q)$ vertex-oriented Hamilton cycle in $D$ for the cases when $|P| \geq {2\over 3}n$ and when ${1\over 3}n \leq |P| \leq {2\over 3}n$. This sharpens a bound given by Badheka et al.

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A counterexample to a conjecture of grant

Cited by 9 publications

References 0 publications

Arbitrary Orientations of Hamilton Cycles in Digraphs

Arbitrary Orientations of Hamilton Cycles in Digraphs

Semi-degree threshold for anti-directed Hamiltonian cycles

Vertex-Oriented Hamilton Cycles in Directed Graphs

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