Proving that every graph is an induced subgraph of an r-regular graph

[GroupTheory1]

Homework Statement

How would you prove that every graph G is an induced subgraph of an r-regular graph where r >= D (the largest degree of the vertices of G)?

Homework Equations

NA

The Attempt at a Solution

I can picture the answer for when G itself could be turned into a D-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 into a D-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 r > D. (I am also having trouble with the just language of graph theory and how to write proofs for it if you couldn't tell.)

 

[mighty2000]

I would approach this problem by first defining some key concepts and terms to make sure we are on the same page.

First, let's define what an induced subgraph is. An induced subgraph of a graph G is a subset of vertices and edges of G that form a new graph H, where the vertices of H are a subset of the vertices of G and the edges of H are a subset of the edges of G. In other words, H is a graph that can be obtained from G by removing some vertices and edges.

Next, let's define what a regular graph is. A regular graph is a graph where all vertices have the same degree. In this case, we are looking for an r-regular graph, where r is the degree of all vertices in the graph.

Now, let's look at the problem at hand. We want to prove that every graph G is an induced subgraph of an r-regular graph where r >= D, the largest degree of the vertices of G.

To prove this, we can use the concept of a degree sequence. The degree sequence of a graph is the list of degrees of all vertices in the graph, arranged in non-increasing order. For example, if a graph has degree sequence (3,3,2,2,1), it means there are two vertices with degree 3, two vertices with degree 2, and one vertex with degree 1.

Now, let's assume that the degree sequence of G is (d_1, d_2, ..., d_n), where d_1 >= d_2 >= ... >= d_n. Since D is the largest degree of the vertices of G, we know that d_1 = D.

To construct an r-regular graph containing G as an induced subgraph, we can simply add vertices and edges to G in a specific way. Let's call this new graph H.

First, we add r - D new vertices to H. These vertices will have degree r, making H an r-regular graph.

Next, we connect each of these new vertices to d_1 - D vertices in G. This ensures that the degree of these new vertices in H is r, and that G is an induced subgraph of H.

We continue this process for each of the remaining vertices in G, adding r - d_i new vertices and connecting them to d_i - d_{i+1} vertices in G. This will