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An extreme problem concerning k-arc-cyclic property for a class of tournaments. (English) Zbl 0723.05061
For fixed integers q and k let \(N=N(q,k)\) denote the smallest integer with the following property: if \(T_ n\) is a tournament with \(n\geq N\) nodes in which \(d^+(v)+d^-(u)\leq n-q\) for every arc uv of \(T_ n\), then every arc of \(T_ n\) belongs to a k-cycle. The authors show that \(N(q,4)=5q-7\) for \(q>1\) and they determine N(q,5) to within one when \(q\geq 15\).
MSC:
05C20 Directed graphs (digraphs), tournaments
05C35 Extremal problems in graph theory
05C38 Paths and cycles
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References:
[1] Alspach, B., Cycles of each length in regular tournaments, Canad. math. bull., 10, 283-286, (1967) · Zbl 0148.43602
[2] Zhu, Y.; Tian, F., On the strong path connectivity of a tournament, Sci. sinica, II, 18-28, (1979), Special Issue
[3] Zhu, Y.; Tian, F.; Chen, C.; Zhang, C., Arc-pancyclic property of tournaments under some degree conditions, J. inform. optim. sci., 1, 5, 1-16, (1984) · Zbl 0534.05032
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