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Extremal Problems in Royal Colorings of Graphs
An edge coloring of a graph is a royal -edge coloring of if
the edges of are assigned nonempty subsets of the set
in such a way that the vertex coloring obtained by assigning the union of the
colors of the incident edges of each vertex is a proper vertex coloring. If the
vertex coloring is vertex-distinguishing, then is a strong royal -edge
coloring. The minimum positive integer for which has a strong royal
-edge coloring is the strong royal index of . It has been conjectured
that if is a connected graph of order where for a positive integer , then the strong royal index of is either
or . We discuss this conjecture along with other information
concerning strong royal colorings of graphs. A sufficient condition for such a
graph to have a strong royal index is presented