ST-COLORING OF JOIN AND DISJOINT UNION OF GRAPHS

R. Moran, N. Bora, A. K. Baruah, and A. Bharali

  DOI:  https://doi.org/10.37418/amsj.9.11.44

Full Text

A Strong $T$-coloring (ST-coloring) of a graph $G=(V,E)$ is a function $c:V(G)\rightarrow Z^+\bigcup \{0\}$ such that for all $u\neq w$ in $V(G)$, if $(u,w)\in E(G)$ then $\mid c(u)-c(w)\mid \notin T$ and $\mid c(u)-c(w)\mid\neq\mid c(x)-c(y)\mid$ for any two distinct edges $(u,w)$ and $(x,y)$ in $E(G)$. For a $ST$-coloring $c$ of the graph G, $c_{ST}-span, sp^{c}_{ST} (G)$ is the maximum value of $\mid c(u)-c(w)\mid$ over all the vertices of G and the minimum of $sp^{c}_{ST} (G)$ is denoted by $sp_{ST}(G)$, where the minimum is taken over all ST-coloring $c$ of $G$. Considering the edges, the $c_{ST}-edgespan, esp^{c}_{ST} (G)$ is the maximum value $\mid c(u)-c(w)\mid$ over all the edges$(u,w)$ and the minimum of $esp^{c}_{ST} (G)$ is defined as $esp_{ST}(G)$, where the minimum is taken over all ST-coloring $c$ of G. In this paper, we establish some results related to $ST$-chromatic number, span and edge span of join and disjoint union of Graphs.

Keywords: ST-coloring, ST-chromatic number, ST-span, ST-edge span.