This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
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  and Bondy and Thomassen . The second proof is slightly simpler than the first one and can also be found in the book  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  proved that by weakening the degree condition in Meyniel’s theorem only by one, we obtain a stronger result: Theorem 5.6.8  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  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  is based on the multi-insertion technique. Theorem 5.6.9  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 . Theorem 5.6.10  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  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 . Theorem 5.6.11  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)  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....
View Full Document
This document was uploaded on 08/10/2011.
- Spring '11