Consider a graph $G(V,E)$. For any two vertices $v_1 , v_2 \in V$,  the subgraph induced by the neigbourhood of vertex $v_1$, denoted by $G_{v_1}$ be isomorphic to the subgraph induced by the neigbourhood of veretx $v_2$, denoted by $G_{v_2}$.Now suppose one can color $G_{v_1}$(and $G_{v_2}$) using $k$ colors. Can one upper-bound the chromatic number of the graph $G$ in terms of $k$ (and other graph EDIT 1: The motivation behind this question is the fact that the chromatic number is upper bounded by max degree of the graph. Therefore it might happen that if the neighbourhood satisfies certain properties (like densely connected?,lot of cycles of all lengths?), may be one can related its chromatic number of graph $G$ to $k$.Links to related literature will be helpful.Thanks in advance!
Chromatic number of a graph where neighbourhood of each vertex is isomorphic
reference request;graph theory;graph isomorphism;graph colouring