Extremal problems in royal colorings of graphs

Abstract

An edge coloring \( c \) of a graph \( G \) is a royal \( k \)-edge coloring of \( G \) if the edges of \( G \) are assigned nonempty subsets of the set \( \{1,2,\dots,k\} \) 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 \( c \) is a strong royal \( k \)-edge coloring. The minimum positive integer \( k \) for which \( G \) has a strong royal \( k \)-edge coloring is the strong royal index of \( G \). It has been conjectured that if \( G \) is a connected graph of order \( n \geq 4 \) where \( 2^{k-1} \leq n \leq 2^k – 1 \) for a positive integer \( k \), then the strong royal index of \( G \) is either \( k \) or \( k+1 \). We discuss this conjecture along with other information concerning strong royal colorings of graphs. A sufficient condition for such a graph to have strong royal index \( k+1 \) is presented.