SOME RESULTS ON $T$-COLORING AND $ST$-COLORING OF GENERALIZED BUTTERFLY GRAPHS

Abstract

Consider a finite set $T$ of positive integers including zero. Then the graph $G$ yields a $T$-coloring if there exists a function $\gamma$ defined on the set of vertices $V(G)$ such that for any edge $(a, b)$ of $G$ and $a \neq b$, $|\gamma(a)-\gamma(b)| \notin T$. A graph $G$ admits a particular type of $T$-coloring, namely, strong $T$-coloring which is defined as the map $\phi$ on the set of vertices so that $\forall a \neq b \in V(G)$, if $(a, b) \in E(G)$, then $|\phi(a)-\phi(b)|$ does not lie in the set $T$ and the values $|\phi(a)-\phi(b)|$ and $|\phi(l)-\phi(m)|$ are distinct for any two distinct edges $(a, b)$ and $(l, m)$ of $G$. ST-chromatic number of the graph $G$, denoted by $\chi_{S T}(G)$ is the least number of colors required for an $S T$-coloring of G. Again, $s p_{S T}(G)$ is the $\min _\phi\{\max |\phi(a)-\phi(b)|\}$ considering each vertex, whereas $\operatorname{esp}_{S T}(G)$ is the $\min _\phi\{\max |\phi(a)-\phi(b)|\}$ considering each edge $(a, b)$ in $G$. In this paper, some results related to chromatic number, span and edge span associated with $T$ - and $S T$-coloring of generalized butterfly graphs are presented.

Received: December 9, 2021
Revised: March 29, 2022
Accepted: April 12, 2022