Problems that are NP-hard to approximate even when the input graph is regular Are there graph problems which are NP-hard to approximate even when the input graph is regular?

For instance, are there optimization problems that are NP-hard to approximate within $O(n^\epsilon)$ even when the input graph is regular?

PS: The input graph is a simple undirected graphs
For all $k\geq 3$, the problem maximum induced matching is APX-hard for $k$-regular bipartite graphs. See this paper. 

A matching $M$ of a graph $G$ is induced if for every pair of edges $e,e'$ in M, there is no edge between them in $G$.