Let $G$ be a regular simple graph with degree $\Delta=n-k-1$ and order $m$. Let $C_k$ be the regular graph which is formed by removing a $k$-factor from the complete graph $K_{n}$. I think we could always find a proper induced subgraph of $C_k$ with maximum degree at least $\ge\frac{\Delta}{2}$ as a subgraph of the graph $G$. Is this true?

If this be true, then I think to find the invariants of $G$, it suffices to find the invariants of $C_k$. Then, the invariants of $G$ would be related linearly with that of $C_k$. For example, the chromatic number/ index seems to be closely related to the number of disjoint copies of $C_k$ which occurs as an induced graph and $m$. In a way, I think this could be related to the tree-decomposition of the graph $G$. Any light on this? Thanks beforehand.

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