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Hamiltonian Cycles Avoiding Prescribed Arcs in Tournaments

Published online by Cambridge University Press:  01 September 1997

JØRGEN BANG-JENSEN
Affiliation:
Department of Mathematics and Computer Science, Odense University, Odense, Denmark (e-mail: jbj@imada.ou.dk, z.g.gutin@brunel.ac.uk, gyeo@imada.ou.dk)
GREGORY GUTIN
Affiliation:
Department of Mathematics and Computer Science, Odense University, Odense, Denmark (e-mail: jbj@imada.ou.dk, z.g.gutin@brunel.ac.uk, gyeo@imada.ou.dk)
ANDERS YEO
Affiliation:
Department of Mathematics and Computer Science, Odense University, Odense, Denmark (e-mail: jbj@imada.ou.dk, z.g.gutin@brunel.ac.uk, gyeo@imada.ou.dk)

Abstract

Thomassen [6] conjectured that if I is a set of k−1 arcs in a k-strong tournament T, then TI has a Hamiltonian cycle. This conjecture was proved by Fraisse and Thomassen [3]. We prove the following stronger result. Let T=(V, A) be a k-strong tournament on n vertices and let X1, X2, [ctdot ], Xl be a partition of the vertex set V of T such that [mid ]X1[mid ][les ][mid ]X2[mid ] [les ][ctdot ][les ][mid ]Xl[mid ]. If k[ges ][sum ] l−1i=1[lfloor ] [mid ]Xi[mid ]/2[rfloor ]+[mid ]Xl[mid ], then T−∪li=1 {xyA[ratio ]x, yXi} has a Hamiltonian cycle. The bound on k is sharp.

Type
Research Article
Copyright
1997 Cambridge University Press

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