TITLE: Proving that every graph is an induced subgraph of an r-regular graph
QUESTION [5 upvotes]: How would you prove that every graph $G$ is an induced subgraph of the $r$-regular graph, where $r\geq \Delta(G)?$
I can picture the answer for when $G$ itself can be turned into a $\Delta$-regular graph: make a union of $G$ with a copy of itself and then connect the vertices across the two vertex sets $U$ (from $G$) and $W$ (from the copy of $G$) such that $u_i$ and $w_j$ are connected if and only if $v_i$ and $v_j$ would be connected in the original graph in order to turn it (the original graph) into a $\Delta$-regular graph.
However, I cannot figure out how to do it in the general case where, for instance, the order of $G$ may be even or odd (and, thus, may not be made into an $r$-regular graph if r is odd as well) or for when $r>\Delta$. (I am also having trouble with the just language of graph theory and how to write proofs for it if you couldn't tell.)

REPLY [5 votes]: Take two copies $G_1$ and $G_2$ of $G$ and add and edge between each vertex $v$ of $G_1$ and every vertex of $G_2$ corresponding to non-neighbours of $v$. Then $G$ is obviously an induced subgraph, the obtained graph is $|G|-1$ regular and has order $2|G|$.