53 research outputs found

    A Note on Long non-Hamiltonian Cycles in One Class of Digraphs

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    Let DD be a strong digraph on n≥4n\geq 4 vertices. In [3, Discrete Applied Math., 95 (1999) 77-87)], J. Bang-Jensen, Y. Guo and A. Yeo proved the following theorem: if (*) d(x)+d(y)≥2n−1d(x)+d(y)\geq 2n-1 and min{d+(x)+d−(y),d−(x)+d+(y)}≥n−1min \{d^+(x)+ d^-(y),d^-(x)+ d^+(y)\}\geq n-1 for every pair of non-adjacent vertices x,yx, y with a common in-neighbour or a common out-neighbour, then DD is hamiltonian. In this note we show that: if DD is not directed cycle and satisfies the condition (*), then DD contains a cycle of length n−1n-1 or n−2n-2.Comment: 7 pages. arXiv admin note: substantial text overlap with arXiv:1207.564

    Generalizations of tournaments: A survey

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    A sufficient condition for a balanced bipartite digraph to be hamiltonian

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    We describe a new type of sufficient condition for a balanced bipartite digraph to be hamiltonian. Let DD be a balanced bipartite digraph and x,yx,y be distinct vertices in DD. {x,y}\{x, y\} dominates a vertex zz if x→zx\rightarrow z and y→zy\rightarrow z; in this case, we call the pair {x,y}\{x, y\} dominating. In this paper, we prove that a strong balanced bipartite digraph DD on 2a2a vertices contains a hamiltonian cycle if, for every dominating pair of vertices {x,y}\{x, y\}, either d(x)≥2a−1d(x)\ge 2a-1 and d(y)≥a+1d(y)\ge a+1 or d(x)≥a+1d(x)\ge a+1 and d(y)≥2a−1d(y)\ge 2a-1. The lower bound in the result is sharp.Comment: 12 pages, 3 figure
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