Let $G$ be any simple graph (i.e has no loops nor multiple edges) and let $1,2,...,\chi(G)$ be any good coloring to the vertices of $G$(i.e a minimal coloring for its vertices in which each 2 adjacent vertices have different colors).
let $v$ be a vertex in $G$ of color $1$ such that $v$ has no neighbors of color $2$.
show that $\chi(G-\{v\})=\chi(G)$.
Thank you for helping.
The problem is false, Indeed, Consider a cycle of 5 vertices colored by the colors 1,2,1,2,3 then the third vertex colored by color 1 and not adjacent to the color 3, but removing it obtain a path which can be colored by 2 colors.