Summary: FACTOR d-DOMATIC COLORINGS OF GRAPHS
NOGA ALON, GUILLAUME FERTIN, ARTHUR L. LIESTMAN, THOMAS C. SHERMER,
AND LADISLAV STACHO
Abstract. Consider a graph and a collection of (not necessarily edge-disjoint) con-
nected spanning subgraphs (factors) of the graph. We consider the problem of coloring
the vertices of the graph so that each color class of the vertices dominates each factor.
We find upper and lower bounds on (t, k), which we define as the minimum radius
of domination d such that every graph with a collection of k factors can be vertex
colored with t colors so that each color class d-dominates each factor. It is perhaps
surprising that the upper bound is finite and does not depend on the order of the
graph. We obtain similar results for a variant of the problem where the number of
colors is equal to the number of factors and each color class must d-dominate only the
corresponding factor rather than all factors.
Factor domination was introduced by Brigham and Dutton [?]. In their paper, a
decomposition of a graph into k edge-disjoint spanning factors is called a k-factoring.
A subset of the vertices of the graph is a factor dominating set if it is a dominating
set for each of the factors. Brigham and Dutton studied the minimum cardinality
of factor dominating sets. A similar concept, global domination, was introduced by
Sampathkumar [?]. A summary of results in this area appears in a paper by Brigham