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Unformatted text preview: 5.7 Longest Paths and Cycles in Semicomplete Multipartite Digraphs 263 Corollary 5.7.25 [744] If a kstrong semicomplete multipartite digraph D has at most k vertices in each partite set, then D contains a Hamilton cycle. ut Corollary 5.7.26 [744] A kstrong semicomplete multipartite digraph has a cycle through any set of k vertices. ut Theorem 5.7.23 was generalized by Yeo [748] as follows (its proof also uses Theorem 5.7.21). Let i l ( D ) = max { d + ( x ) d ( x )  : x ∈ V ( D ) } and i g ( D ) = Δ ( D ) δ ( D ) for a digraph D (the two parameters are called the local irregularity and the global irregularity , respectively, of D [748]). Clearly, i l ( D ) ≤ i g ( D ) for every digraph D . Theorem 5.7.27 [748] Let D be a semicomplete cpartite digraph of order n with partite sets of cardinalities n 1 ,n 2 ,...,n c such that n 1 ≤ n 2 ≤ ... ≤ n c . If i g ( D ) ≤ ( n n c 1 2 n c ) / 2 + 1 or i l ( D ) ≤ min { n 3 n c + 1 , ( n n c 1 2 n c ) / 2 + 1 } , then D is hamiltonian. ut The result of this theorem is best possible in a sense: Yeo [748] constructed an infinite family D of nonhamiltonian semicomplete multipartite digraphs such that every D ∈ D has i l ( D ) = i g ( D ) = ( n n c 1 2 n c + 1) / 2 + 1 ≤ n 3 n c + 2. Another generalization of Theorem 5.7.23, whose proof is based on The orem 5.7.21, was obtained by Guo, Tewes, Volkmann and Yeo [348]. For a digraph D and a positive integer k , define f ( D,k ) = X x ∈ V ( D ) ,d + ( x ) >k ( d + ( x ) k ) + X x ∈ V ( D ) ,d ( x ) <k ( k d ( x )) . Theorem 7.5.3 in Ore’s book [595] on the existence of a perfect matching in a bipartite graph can easily be transformed into a sufficient condition for a digraph to contain a cycle factor. This condition is as follows. If, for a digraph D and positive integer k , we have f ( D,k ) ≤ k 1, then D has a cycle factor. For a positive integer k ≥ 2, let G k be a semicomplete 3partite digraph with the partite sets V 1 = { x } , V 2 = { y 1 ,y 2 ,...,y k 1 } , and V 3 = { z 1 ,z 2 ,...,z k } and arc set { yx,xz,zy,yv : y ∈ V 2 ,z ∈ V 3 ,v ∈ V 3 z 1 } ∪ { z 1 x } . The digraph G 00 k is the converse of G k . We observe that f ( G k ,k ) = k 1 (Exercise 5.43), but G k is not hamiltonian, as a hamiltonian cycle would contain the arc xz 1 and every second vertex on the cycle would belong to the partite set V 3 . Since x has no inneighbour in V 3 z 1 , this is not possible. Clearly, G 00 k is not hamiltonian either. Theorem 5.7.28 [348] Let D be a semicomplete multipartite digraph such that f ( D,k ) ≤ k 1 for some positive integer k . If D is not isomorphic to G k or G 00 k , then D is Hamiltonian. ut 264 5. Hamiltonicity and Related Problems The authors of [348] introduced the following family of semicomplete mul tipartite digraphs. Let D be a semicomplete multipartite digraph with par tite sets V 1 ,V 2 , ...,V k . If min { ( x i ,V j )  ,  ( V j ,x i ) } ≥ 1 2  V j  for every ver tex...
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This document was uploaded on 08/10/2011.
 Spring '11
 Algorithms

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