<i>k</i>-tuple total dominator chromatic number and Mycielskian graphs

Abstract The k -tuple total dominator chromatic number is a graph parameter that measures the minimum number of colors required for a proper coloring, where each vertex must be adjacent to every vertex of k distinct color classes. In this paper, we investigate the k -tuple total dominator chromatic number in the context of Mycielskian graphs. Mycielskian graphs are obtained through a graph transformation introduced by Mycielski in 1955, and they have been widely studied for their intriguing properties and applications in graph theory. The generalized Mycielskians are natural generalizations of Mycielskian graphs, denoted by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>μ</m:mi> <m:mi>m</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mu_{m}(G)} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {m\geq 1} . Specifically, we determine both the upper and lower bounds for the k -tuple total dominator chromatic number of Mycielskian graphs, providing insights into the minimum number of colors needed to achieve a k -tuple total dominator coloring in these graphs. We also discuss the exact values of the k -tuple total dominator chromatic number for specific instances of Mycielskian graphs. Moreover, we discuss the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>χ</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi>d</m:mi> </m:mrow> <m:mi>t</m:mi> </m:msubsup> </m:math> {\chi_{k,d}^{t}} of the generalized Mycielskian graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>μ</m:mi> <m:mi>m</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {\mu_{m}(G)} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>m</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {m\geq 2} , by giving some bounds in terms of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi>m</m:mi> </m:mrow> </m:math> {k,m} and that of G .