Neighborhood Total Domination and Colouring in Graphs

Let $G=(V,E)$ be a graph without isolated vertices. A dominating set $S$ of $G$ is a neighborhood total dominating set (ntd-set) if the induced subgraph $leftlangle N(S)rightrangle$ of $G$ has no isolated vertices. The neighborhood total domination number $gamma_{nt}(G)$ is the minimum cardinality of a ntd-set. The minimum number of colours required to colour all the vertices such that no two adjacent vertices have same colour is the chromatic number $chi(G)$ of $G$. In this paper we find an upper bound for sum of the ntd-number and chromatic number and characterize the corresponding extremal graphs.