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Unformatted text preview: 5.6 Hamilton Cycles and Paths in DegreeConstrained Digraphs 243 Short proofs of Meyniel’s theorem were given by OverbeckLarisch [597] and Bondy and Thomassen [128]. The second proof is slightly simpler than the first one and can also be found in the book [735] by West (see Theorem 8.4.38). Using Proposition 5.0.3 one can easily see that replacing 2 n 1 by 2 n 3 in Meyniel’s theorem we obtain sufficient conditions for traceability. (Note that for traceability we do not require strong connectivity.) Darbinyan [180] proved that by weakening the degree condition in Meyniel’s theorem only by one, we obtain a stronger result: Theorem 5.6.8 [180] Let D be a digraph of order n ≥ 3 . If d ( x ) + d ( y ) ≥ 2 n 2 for all pairs of nonadjacent vertices in D , then D contains a hamil tonian path in which the initial vertex dominates the terminal vertex. ut Berman and Liu [111] extended Theorem 5.6.7 as formulated below. For a digraph D of order n , a set M ⊆ V ( D ) is Meyniel if d ( x ) + d ( y ) ≥ 2 n 1 for every pair x,y of nonadjacent vertices in M . The proof of Theorem 5.6.9 in [111] is based on the multiinsertion technique. Theorem 5.6.9 [111] Let M be a Meyniel set of vertices of a strong digraph D of order n ≥ 2 . Then D has a cycle containing all vertices of M . ut Another extension of Meyniel’s theorem was given by Heydemann [428]. Theorem 5.6.10 [428] Let h be a nonnegative integer and let D be a strong digraph of order n ≥ 2 such that, for every pair of nonadjacent vertices x and y , we have d ( x ) + d ( y ) ≥ 2 n 2 h +1 . Then D contains a cycle of length greater than or equal to d n 1 h +1 e + 1 . ut Manoussakis [547] proved the following sufficient condition that involves triples rather than pairs of vertices. Notice that Theorem 5.6.11 does not imply either of Theorems 5.6.1, 5.6.5 and 5.6.7 [69]. Theorem 5.6.11 [547] Suppose that a strong digraph D of order n ≥ 2 satisfies the following conditions: for every triple x,y,z ∈ V ( D ) such that x and y are nonadjacent (a) If there is no arc from x to z , then d ( x )+ d ( y )+ d + ( x )+ d ( z ) ≥ 3 n 2 . (b) If there is no arc from z to x, then d ( x )+ d ( y )+ d ( x )+ d + ( z ) ≥ 3 n 2 . Then D is hamiltonian. ut The next theorem resembles both Theorem 5.6.5 and Theorem 5.6.7. How ever, Theorem 5.6.12 does not imply any of these theorems. The sharpness of the inequality of Theorem 5.6.12 can be seen from the digraph H n introduced before Theorem 5.6.7. 244 5. Hamiltonicity and Related Problems Theorem 5.6.12 (Zhao and Meng) [758] Let D be a strong digraph of order n ≥ 2 . If d + ( x ) + d + ( y ) + d ( u ) + d ( v ) ≥ 2 n 1 for every pair x,y of dominating vertices and every pair u,v of dominated vertices, then D is hamiltonian. ut Theorems 5.6.5 and 5.6.12 suggest that the following conjecture by Bang Jensen, Gutin and Li, may be true....
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This document was uploaded on 08/10/2011.
 Spring '11
 Algorithms

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