How would you prove that every graph G$G$ is an induced subgraph of an rthe $r$-regular graph, where r >= D and D is the largest degree of the vertices of G?$r\geq \Delta(G)?$
I can picture the answer for when G$G$ itself can be turned into a D$\Delta$-regular graph: make a union of G$G$ with a copy of itself and then connect the vertices across the two vertex sets U $U$ (from G$G$) and W $W$ (from the copy of G$G$) such that u_i$u_i$ and w_j$w_j$ are connected if and only if v_i$v_i$ and v_j$v_j$ would be connected in the original graph in order to turn it (the original graph) into a D$\Delta$-regular graph.
However, I cannot figure out how to do it in the general case where, for instance, the order of G$G$ may be even or odd (and, thus, may not be made into an r$r$-regular graph if r is odd as well) or for when r > D$r>\Delta$. (I am also having trouble with the just language of graph theory and how to write proofs for it if you couldn't tell.)
