If $G=(V,E)$ is a simple, undirected graph, is there a regular graph $G\_R$ such that $G$ is isomorphic to an induced subgraph of $G\_R$ and $\chi(G) = \chi(G\_R)$?



Embedding any graph in a regular graph with the same chromatic number
Yes, you can find such a $G\_R$ of any degree greater than or equal to the maximum degree of $G$. This is the main theorem of the paper ["On regular bipartite-preserving supergraphs"](https://link.springer.com/article/10.1007%2FBF01834121) by G. Chartrand and C. E. Wall .


