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Digraphs+Theory,+Algorithms+and+Applications_Part15

# Digraphs+Theory,+Algorithms+and+Applications_Part15 - 5.7...

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5.7 Longest Paths and Cycles in Semicomplete Multipartite Digraphs 263 Corollary 5.7.25 [744] If a k -strong 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 k -strong 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 0 ( D ) - δ 0 ( 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 c -partite 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 non-hamiltonian 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 0 k be a semicomplete 3-partite 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 0 k . We observe that f ( G 0 k , k ) = k - 1 (Exercise 5.43), but G 0 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 in-neighbour 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 0 k or G 00 k , then D is Hamiltonian. ut

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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 x i V i and for every 1 i, j k, j 6 = i , then D is called a semi-partitioncomplete digraph . Several sufficient conditions to guar- antee hamiltonicity of semi-partitioncomplete digraphs were derived in [348].
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