2 research outputs found
Dominated and dominator colorings over (edge) corona and hierarchical products
Dominator coloring of a graph is a proper (vertex) coloring with the property
that every vertex is either alone in its color class or adjacent to all
vertices of at least one color class. A dominated coloring of a graph is a
proper coloring such that every color class is dominated with at least one
vertex. The dominator chromatic number of corona products and of edge corona
products is determined. Sharp lower and upper bounds are given for the
dominated chromatic number of edge corona products. The dominator chromatic
number of hierarchical products is bounded from above and the dominated
chromatic number of hierarchical products with two factors determined. An
application of dominated colorings in genetic networks is also proposed
Dominator Colorings of Digraphs
This paper serves as the first extension of the topic of dominator colorings
of graphs to the setting of digraphs. We establish the dominator chromatic
number over all possible orientations of paths and cycles. In this endeavor we
discover that there are infinitely many counterexamples of a graph and subgraph
pair for which the subgraph has a larger dominator chromatic number than the
larger graph into which it embeds. Finally, a new graph invariant measuring the
difference between the dominator chromatic number of a graph and the chromatic
number of that graph is established and studied. The paper concludes with some
of the possible avenues for extending this line of research.Comment: 23 page