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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
##### Keywords:
cycles; degrees; tournament
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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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