Let $G$ be a regular connected simple graph on $n$ vertices with chromatic number $\chi$ and maximum degree $\Delta$. Then, it is implied that $G$ is $\chi$-partite. Suppose, we remove one of the partite set of vertices. Then, what would be the maximum degree of the induced subgraph formed by the remaining vertices?

I may say with some confidence that the induced subgraph would have a maximum degree of $\textit{at least} $ $\chi-2$(as the remaining partite sets must be connected with each other, otherwise the graph would be disconnected). In addition, if the graph be vertex transitive, I think that the maximum degree of the induced subgraph would be $\Delta-1$. Any hints and counterexamples in this case? Thanks beforehand.

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