Discussiones Mathematicae Graph Theory 30(3) (2010) 385-391
doi: 10.7151/dmgt.1501

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Rangaswami Balakrishnan  and  T. Kavaskar

Srinivasa Ramanujan Centre
SASTRA University
Kumbakonam - 612 001, India
e-mail: mathbala@satyam.net.in
e-mail: t_kavaskar@yahoo.com


A fall coloring of a graph G is a proper coloring of the vertex set of G such that every vertex of G is a color dominating vertex in G (that is, it has at least one neighbor in each of the other color classes). The fall coloring number χf(G) of G is the minimum size of a fall color partition of G (when it exists). Trivially, for any graph G, χ(G) ≤ χf(G). In this paper, we show the existence of an infinite family of graphs G with prescribed values for χ(G) and χf(G). We also obtain the smallest non-fall colorable graphs with a given minimum degree δ and determine their number. These answer two of the questions raised by Dunbar et al.

Keywords: fall coloring of graphs, non-fall colorable graphs.

2010 Mathematics Subject Classification: 05C15.


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Received 18 March 2009
Revised 27 July 2009
Accepted 17 August 2009