Abstract
Gallai and Milgram have shown that the vertices of a directed graph, with stability number α(G), can be covered by exactly α(G) disjoint paths. However, the various proofs of this result do not imply the existence of a maximum stable setS and of a partition of the vertex-set into paths μ1, μ2, ..., μk such tht |μi ∩S|=1 for alli.
Later, Gallai proved that in a directed graph, the maximum number of vertices in a path is at least equal to the chromatic number; here again, we do not know if there exists an optimal coloring (S 1,S 2, ...,S k) and a path μ such that |μ ∩S i|=1 for alli.
In this paper we show that many directed graphs, like the perfect graphs, have stronger properties: for every maximal stable setS there exists a partition of the vertex set into paths which meet the stable set in only one point. Also: for every optimal coloring there exists a path which meets each color class in only one point. This suggests several conjecties similar to the perfect graph conjecture.
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References
D. Amar, I. Fournier andA. Germa,Private communication, May 1981.
J. C. Bermond,Private communication, February 1978.
J. A. Bondy, Disconnected orientations and a conjecture of Las Vergnas,J. London Math. Soc. 2, 14, (1976), 277–282.
J. A. Bondy andU. S. R. Murty,Graph Theory with Applications, MacMillan, London, 1972.
P. Camion, Chemins et circuits des graphes complets,C. R. Acad. Sci. Paris 249 (1959), 2151–52.
T. Gallai, On directed Paths and Circuits, in:Theory of Graphs, (eds. P. Erdős and G. Katona) Academic Press, New York, 1968, 115–118.
T. Gallai andA. N. Milgram, Verallgemeinerung eines Graphentheoretischen Satzes von Rédei,Acta Sc. Math. 21 (1960) 181–186.
M. Las Vergnas, Sur les arborescences dans un graphe orienté,Discrete Math. 15 (1976), 27–29.
M. Las Vergnas, Sur les circuits dans les sommes complètées de Graphes orientés, Institut de Hautes Etudes de Belgique,Colloque sur la Théorie des Graphes, 1973, 231–244.
N. Linial, Covering Digraphs by Paths,Discr. Math. 23 (1978), 257–272.
H. Meyniel,Private communication, May 1981.
L. Pósa, On the circuits of finite graphs,Publ. Math. Inst. Hung. Ac. Sc. 8 (1963) 355–361.
B. Roy, Nombre chromatique et plus longs chemins,Rev. Fr. Automat. Informat. 1 (1967), 127–132.
A. Sache,La Théorie des Graphes, Presses Universitaires de France1554, Paris 1974.
D. De Werra, On the existence of generalized good and equitable edge colorings,J. Graph Theory 5, no 1 (to appear).
V. Chvátal andJ. Komlós, Some Combinatorial Theorems on Monotonity,Canad. Math. Bull.,14 (2), (1971).
P. Erdős andG. Szekeres, A Combinatorial problem in geometry,Compositio Math. 2, (1935), 463–470.
C. C. Chen andP. Manalastas, Every strongly connected digraph of stability 2 has a Hamilton path.To appear in Discrete Math.
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Dedicated to Tibor Gallai on his seventieth birthday