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Coloring, location and domination of corona graphs
A vertex coloring of a graph is an assignment of colors to the vertices
of such that every two adjacent vertices of have different colors. A
coloring related property of a graphs is also an assignment of colors or labels
to the vertices of a graph, in which the process of labeling is done according
to an extra condition. A set of vertices of a graph is a dominating set
in if every vertex outside of is adjacent to at least one vertex
belonging to . A domination parameter of is related to those structures
of a graph satisfying some domination property together with other conditions
on the vertices of . In this article we study several mathematical
properties related to coloring, domination and location of corona graphs.
We investigate the distance- colorings of corona graphs. Particularly, we
obtain tight bounds for the distance-2 chromatic number and distance-3
chromatic number of corona graphs, throughout some relationships between the
distance- chromatic number of corona graphs and the distance- chromatic
number of its factors. Moreover, we give the exact value of the distance-
chromatic number of the corona of a path and an arbitrary graph. On the other
hand, we obtain bounds for the Roman dominating number and the
locating-domination number of corona graphs. We give closed formulaes for the
-domination number, the distance- domination number, the independence
domination number, the domatic number and the idomatic number of corona graphs.Comment: 18 page
On the metric dimension of corona product graphs
Given a set of vertices of a connected graph , the
metric representation of a vertex of with respect to is the vector
, where ,
denotes the distance between and . is a resolving set for if
for every pair of vertices of , . The metric
dimension of , , is the minimum cardinality of any resolving set for
. Let and be two graphs of order and , respectively. The
corona product is defined as the graph obtained from and by
taking one copy of and copies of and joining by an edge each
vertex from the -copy of with the -vertex of . For any
integer , we define the graph recursively from
as . We give several results on the metric
dimension of . For instance, we show that given two connected
graphs and of order and , respectively, if the
diameter of is at most two, then .
Moreover, if and the diameter of is greater than five or is
a cycle graph, then $dim(G\odot^k H)=n_1(n_2+1)^{k-1}dim(K_1\odot H).
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