Improved Sufficient Conditions for the Existence of Anti-Directed Hamiltonian Cycles in Digraphs

Busch, Arthur H.; Jacobson, Michael S.; Morris, Tim R.; Plantholt, Michael J.; Tipnis, Shailesh K.

doi:10.1007/s00373-011-1116-0

Cited by 5 publications

(2 citation statements)

References 11 publications

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“…Proof of Claim 3.7. Let n and m be as in (1). Let P be a maximum length proper ADP in D − R containing P * as an initial segment.…”

Section: Coveringmentioning

confidence: 99%

“…Then in 2008, Plantholt and Tipnis improved upon Grant's result by showing that if D is a directed graph on 2n vertices and δ 0 (D) ≥ 4 3 n, then D has an ADHC [15] (note that this is for all n). Finally in 2011, Busch, Jacobson, Morris, Plantholt, Tipnis improved upon all the results for ADHC's by showing that if D is a directed graph on 2n vertices and δ 0 (D) ≥ n + 14 3 √ n, then D has an ADHC [1].…”

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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“…Proof of Claim 3.7. Let n and m be as in (1). Let P be a maximum length proper ADP in D − R containing P * as an initial segment.…”

Section: Coveringmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 96%