Digraphs+Theory,+Algorithms+and+Applications_Part14

Digraphs+Theory,+Algorithms+and+Applications_Part14 - 5.6...

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Unformatted text preview: 5.6 Hamilton Cycles and Paths in Degree-Constrained Digraphs 243 Short proofs of Meyniel’s theorem were given by Overbeck-Larisch [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 non-adjacent 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 non-adjacent vertices in M . The proof of Theorem 5.6.9 in [111] is based on the multi-insertion 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 non-negative integer and let D be a strong digraph of order n ≥ 2 such that, for every pair of non-adjacent 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 non-adjacent (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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Digraphs+Theory,+Algorithms+and+Applications_Part14 - 5.6...

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