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Unformatted text preview: 6.5 Pancyclicity of Digraphs 303 ( i = 0 , 1 , 2) contains a directed path of length 2, then D contains no cycle of length 5. Now we prove the sufficiency of the condition in (a). According to Theo rem 4.8.5, there exists a semicomplete digraph T on k vertices for some k ≥ 3 such that D is obtained from T by substituting a quasitransitive digraph H v for each vertex v ∈ V ( T ) (here H v is nonstrong if it has more than one vertex). Let C be a hamiltonian cycle of D . We construct an extended semi complete digraph D from D in the following way: for each H v ,v ∈ V ( T ), first pathcontract each maximal subpath of C which is contained in H v and then delete the remaining arcs of H v . In this process C is changed to a hamiltonian cycle C of D . Suppose D is not pancyclic. Then it is easy to see that D is not pancyclic. By Theorem 6.5.6, D is triangular with a partition V ,V 1 ,V 2 . Let V i ⊂ V be obtained from V i , i = 0 , 1 , 2, by substituting back all vertices on contracted subpaths of C . Then D is triangular with partition V ,V 1 ,V 2 . Moreover each D h V i i is covered by r disjoint subpaths of C for some r . By Lemma 6.5.7, two of V ,V 1 ,V 2 , say V 1 and V 2 , induce subdigraphs with no arcs in D . If  V  =  V 1  =  V 2  we have the first exception in (a). Hence we may assume that  V  >  V 1  =  V 2  . Then D h V i contains an arc of C . From Lemma 6.5.8, we see that D h V i contains no path of length 2. This completes the proof of (a). The proof of (b) is left to the reader as Exercise 6.25. ut 6.5.3 Pancyclic and VertexPancyclic Locally Semicomplete Digraphs We saw in the last subsection how the structure theorem for quasitransitive digraphs (i.e., Theorem 4.8.5) was helpful in finding a characterization for (vertex)pancyclic quasitransitive digraphs. Now we show that the structure theorem for locally semicomplete digraphs (Theorem 4.11.15) is also very useful for finding a characterization of those locally semicomplete digraphs which are (vertex)pancyclic. Our first goal (Lemma 6.5.13) is a characteri zation of those round decomposable locally semicomplete digraphs which are (vertex)pancyclic. Lemma 6.5.10 Let R be a strong round local tournament and let C be a shortest cycle of R and suppose C has k ≥ 3 vertices. Then for every round labelling v ,v 1 ,...,v n 1 of R such that v ∈ V ( C ) there exist indices < a 1 < a 2 < ... < a k 1 < n so that C = v v a 1 v a 2 ...v a k 1 v . Proof: Let C be a shortest cycle and let L = v ,v 1 ,...,v n 1 be a round labelling of R so that v ∈ V ( C ). If the claim is not true, then there exists a number 2 ≤ l < k 1 so that C = v v a 1 v a 2 ...v a k 1 v , where 0 < a 1 < ... < a l 1 and a l < a l 1 . Now the fact that L is a round labelling of R implies that v l 1 → v , contradicting the fact that C is a shortest cycle. ut 304 6. Hamiltonian Refinements Recall that the girth g ( D ) of a digraph is the length of a shortest cycle in D = ( V,A ). For a vertex)....
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 Spring '11
 Algorithms

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