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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 quasi-transitive digraph H v for each vertex v ∈ V ( T ) (here H v is non-strong 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 path-contract 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 Vertex-Pancyclic Locally Semicomplete Digraphs We saw in the last subsection how the structure theorem for quasi-transitive digraphs (i.e., Theorem 4.8.5) was helpful in finding a characterization for (vertex-)pancyclic quasi-transitive 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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