The B-Domatic Number of a Graph

Abstract Besides the classical chromatic and achromatic numbers of a graph related to minimum or minimal vertex partitions into independent sets, the b-chromatic number was introduced in 1998 thanks to an alternative definition of the minimality of such partitions. When independent sets are replaced by dominating sets, the parameters corresponding to the chromatic and achromatic numbers are the domatic and adomatic numbers d(G) and ad(G). We introduce the b-domatic number bd(G) as the counterpart of the b-chromatic number by giving an alternative definition of the maximality of a partition into dominating sets. We initiate the study of bd(G) by giving some properties and examples.

[1]  Mario Valencia-Pabon,et al.  On Approximating the B-Chromatic Number , 2003, Discret. Appl. Math..

[2]  Mario Valencia-Pabon.  Idomatic partitions of direct products of complete graphs , 2010, Discret. Math..

[3]  Stephen T. Hedetniemi,et al.  Disjoint independent dominating sets in graphs , 1976, Discret. Math..

[4]  Gerard J. Chang,et al.  The domatic number problem , 1994, Discret. Math..

[5]  David F. Manlove.  Minimaximal and maximinimal optimisation problems : a partial order-based approach , 1998 .

[6]  Mekkia Kouider,et al.  Some bounds for the b-chromatic number of a grap , 2002, Discret. Math..

[7]  Chính T. Hoàng,et al.  A Characterization of b‐Perfect Graphs , 2010, J. Graph Theory.

[8]  Fred B. Schneider,et al.  A Theory of Graphs , 1993 .

[9]  David Manlove,et al.  The b-chromatic Number of a Graph , 1999, Discret. Appl. Math..

[10]  Frank Harary,et al.  An interpolation theorem for graphical homomorphisms , 1967 .

[11]  Anja Kohl,et al.  Upper bounds on the b-chromatic number and results for restricted graph classes , 2011, Discuss. Math. Graph Theory.

[12]  Sergio Cabello,et al.  On the b-chromatic number of regular graphs , 2011, Discret. Appl. Math..

[13]  Bohdan Zelinka.  Domatically critical graphs , 1980 .

[14]  Bohdan Zelinka,et al.  Adomatic and idomatic numbers of graphs , 1983 .

[15]  Saeed Shaebani.  On Fall Colorings of Graphs , 2015, Ars Comb..

[16]  Jaroslav Ivanco.  An interpolation theorem for partitions which are indivisible with respect to cohereditary properties , 1991, J. Comb. Theory, Ser. B.

[17]  Stephen T. Hedetniemi,et al.  Towards a theory of domination in graphs , 1977, Networks.

[18]  Zsolt Tuza,et al.  On the b-Chromatic Number of Graphs , 2002, WG.

[19]  S. Hedetniemi,et al.  The achromatic number of a graph , 1970 .