# On a conjecture of quintas and arc-traceability in upset tournaments

Arthur H. Busch; Michael S. Jacobson; K. Brooks Reid

Discussiones Mathematicae Graph Theory (2005)

- Volume: 25, Issue: 3, page 225-239
- ISSN: 2083-5892

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topArthur H. Busch, Michael S. Jacobson, and K. Brooks Reid. "On a conjecture of quintas and arc-traceability in upset tournaments." Discussiones Mathematicae Graph Theory 25.3 (2005): 225-239. <http://eudml.org/doc/270331>.

@article{ArthurH2005,

abstract = {A digraph D = (V,A) is arc-traceable if for each arc xy in A, xy lies on a directed path containing all the vertices of V, i.e., hamiltonian path. We prove a conjecture of Quintas [7]: if D is arc-traceable, then the condensation of D is a directed path. We show that the converse of this conjecture is false by providing an example of an upset tournament which is not arc-traceable. We then give a characterization for upset tournaments in terms of their score sequences, characterize which arcs of an upset tournament lie on a hamiltonian path, and deduce a characterization of arc-traceable upset tournaments.},

author = {Arthur H. Busch, Michael S. Jacobson, K. Brooks Reid},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {tournament; upset tournament; traceable; arc-traceability; Hamiltonian path},

language = {eng},

number = {3},

pages = {225-239},

title = {On a conjecture of quintas and arc-traceability in upset tournaments},

url = {http://eudml.org/doc/270331},

volume = {25},

year = {2005},

}

TY - JOUR

AU - Arthur H. Busch

AU - Michael S. Jacobson

AU - K. Brooks Reid

TI - On a conjecture of quintas and arc-traceability in upset tournaments

JO - Discussiones Mathematicae Graph Theory

PY - 2005

VL - 25

IS - 3

SP - 225

EP - 239

AB - A digraph D = (V,A) is arc-traceable if for each arc xy in A, xy lies on a directed path containing all the vertices of V, i.e., hamiltonian path. We prove a conjecture of Quintas [7]: if D is arc-traceable, then the condensation of D is a directed path. We show that the converse of this conjecture is false by providing an example of an upset tournament which is not arc-traceable. We then give a characterization for upset tournaments in terms of their score sequences, characterize which arcs of an upset tournament lie on a hamiltonian path, and deduce a characterization of arc-traceable upset tournaments.

LA - eng

KW - tournament; upset tournament; traceable; arc-traceability; Hamiltonian path

UR - http://eudml.org/doc/270331

ER -

## References

top- [1] K.T. Balińska, M.L. Gargano and L.V. Quintas, An edge partition problem concerning Hamilton paths, Congr. Numer. 152 (2001) 45-54. Zbl0995.05089
- [2] K.T. Balińska, K.T. Zwierzyński, M.L. Gargano and L.V. Quintas, Graphs with non-traceable edges, Computer Science Center Report No. 485, The Technical University of Poznań (2002). Zbl1046.05038
- [3] K.T. Balińska, K.T. Zwierzyński, M.L. Gargano and L.V. Quintas, Extremal size problems for graphs with non-traceable edges, Congr. Numer. 162 (2003) 59-64. Zbl1046.05038
- [4] J. Bang-Jensen and G. Gutin, Digraphs: Theory, Algorithms and Applications (Springer-Verlag, Berlin, 2001). Zbl0958.05002
- [5] R.A. Brualdi and Q. Li, The interchange graph of tournaments with the same score vector, in: Progress in graph theory, Proceedings of the conference on combinatorics held at the University of Waterloo, J.A. Bondy and U.S.R. Murty editors (Academic Press, Toronto, 1982), 129-151.
- [6] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs (Academic Press, New York, 1980). Zbl0541.05054
- [7] L.V. Quintas, private communication, (2001).
- [8] L. Rédei, Ein Kombinatorischer Satz, Acta Litt. Szeged. 7 (1934) 39-43.
- [9] K.B. Reid, Tournaments, in: The Handbook of Graph Theory, Jonathan Gross and Jay Yellen editors (CRC Press, Boca Raton, 2004).

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