Abstract
A linear directed forest is a directed graph in which every component is a directed path. The linear arboricity la(D) of a digraph D is the minimum number of linear directed forests in D whose union covers all arcs of D. For every d-regular digraph D, Nakayama and Péroche conjecture that la(D) = d + 1. In this paper, we consider the linear arboricity for complete symmetric digraphs, regular digraphs with high directed girth and random regular digraphs and we improve some well-known results. Moreover, we propose a more precise conjecture about the linear arboricity for regular digraphs.
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References
Akiyama, J., Exoo, G., Harary, F.: Covering and packing in graphs IV: linear arboricity. Networks, 11, 69–72 (1981)
Alon, N.: Probabilistic methods in coloring and decomposition problems. Discerte Math., 127, 31–46 (1994)
Alspach, B., Bermond, J. C., Sotteau, D.: Decomposition into cycles I: Hamilton decompositions. In Cycles and Rays, Springer, Netherlands, 1990, 9–18
Bissacot, R., Fernández, R., Procacci, A., et al.: An improvement of the Lovász Local Lemma via cluster expansion. Combin. Probab. Comput., 20(05), 709–719 (2011)
Bondy, J. A., Murty, U. S. R.: Graph Theory with Applications, North-Holland, New York, 1976
Bollobás, B.: Random Graphs, Springer, New York, 1998
Cooper, C., Frieze, A., Molloy, M.: Hamilton cycles in random regular digraphs. Combin. Probab. Comput., 3(01), 39–49 (1994)
Nakayama, A., Péroche, B.: Linear arboricity of digraphs. Networks, 17, 39–53 (1987)
Ndreca, S., Procacci, A., Scoppola, B.: Improved bounds on coloring of graphs. European J. Combin., 33(4), 592–609 (2012)
Tillson, T. W.: A Hamiltonian decomposition of K *2m , 2m = 8. J. Combin. Theory, Ser. B, 29(1), 68–74 (1980)
Wood, D. R.: Bounded degree acyclic decompositions of digraphs. J. Combin. Theory, Ser. B, 90, 309–313 (2004)
Acknowledgements
We wish to thank the referees for their valuable suggestions. Especially, one of the referees told us the definition of linear arboricity, which leads us to rewrite the whole manuscript and improve one of the main results. We really appreciate it.
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Supported by NSFC (Grant Nos. 11601093 and 11671296)
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He, W.H., Li, H., Bai, Y.D. et al. Linear arboricity of regular digraphs. Acta. Math. Sin.-English Ser. 33, 501–508 (2017). https://doi.org/10.1007/s10114-016-5071-9
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DOI: https://doi.org/10.1007/s10114-016-5071-9