Received: 22 September 2016 Revised: 23 August 2017 Accepted: 9 October 2017
Thoroughly dispersed colorings
Michael A. Henning
School of Computing and Department of
Mathematical Sciences, Clemson University,
Clemson, SC 29634, USA
Department of Pure and Applied Mathemat-
ics, University of Johannesburg, Auckland Park
2006, South Africa
WayneGoddard, School of Computing
and Department of Mathematical Sciences,
Clemson University, Clemson, SC 29634, USA.
Contract grant sponsor: South African National
ResearchFoundation and the University of
We consider (not necessarily proper) colorings of the ver-
tices of a graph where every color is thoroughly dispersed,
that is, appears in every open neighborhood. Equivalently,
every color is a total dominating set. We deﬁne td() as
the maximum number of colors in such a coloring and
FTD() as the fractional version thereof. In particular, we
show that every claw-free graph with minimum degree at
least two has FTD() ≥ 3∕2 and this is best possible.
For planar graphs, we show that every triangular disc has
FTD() ≥ 3∕2 and this is best possible, and that every pla-
nar graph has td() ≤ 4 and this is best possible, while we
conjecture that every planar triangulation has td() ≥ 2.
Further, although there are arbitrarily large examples of
connected, cubic graphs with td()=1, we show that for
a connected cubic graph FTD() ≥ 2−(1). We also con-
sider the related concepts in hypergraphs.
coloring, thoroughly dispersed coloring, total domination, transversal
AMS SUBJECT CLASSIFICATION:
We consider (not necessarily proper) colorings of the vertices of a graph where every color is thor-
oughly dispersed, that is, appears in every open neighborhood. Chen et al.  called this the coupon
We deﬁne td() as the maximum number of colors in such a coloring. Note that a color being thor-
oughly dispersed is equivalent to a color being a total dominating set (a set of vertices such that
every vertex has a neighbor in ). Thus the parameter td() is equivalent to the maximum number of
174 © 2017 Wiley Periodicals, Inc. wileyonlinelibrary.com/journal/jgt J Graph Theory. 2018;88:174–191.