I am following "Turan's Graph Theorem" by M. Aigner. In the second proof by Erdos, he states,
"Applying induction on S, we thus infer that among the graphs with a maximal number of edges there is a graph $K_{n_1,…,n_{k-1}}$"
is where I get confused with the proof. What does he mean by induction on S? I feel as I need help with the steps between the construction of H, how it gives the upperbound for edges of G (i.e $|E(H)|\geq|E(G)|$) , and how by "induction on S" we know there's a Turan graph among the graphs with maximal edges (also, does this imply that graph H is that Turan graph-if so, what guarantees it?).
Thank you
Best Answer
In this case, "Applying induction on $S$" means "Applying the $k-1$ case of the theorem to the subgraph $G[S]$".
Formally, the theorem we're proving here is the following version of TurĂ¡n's theorem:
Going through all the things you want help with:
The construction of $H$. We let $m$ be a vertex in $G$ with the largest degree; $S$ is the set of neighbors of $m$, and $T$ is the complement of $S$ (so $m \in T$). To get $H$ from $G$, we:
Why is $|E(H)| \ge |E(G)|$? We prove a stronger statement: for all vertices $v$, $\deg_H(v) \ge \deg_G(v)$. This gives us an inequality between the edge counts, since the sum of degrees gives us twice the number of edges.
We check the degree inequality separately in two cases:
How does the induction work? It is an induction on $k$. The base case can be $k=2$: the only graph with no $2$-cliques (edges) is the empty graph, which is a $1$-partite graph.
If $G$ started out as a graph with no $k$-cliques, then the subgraph $G[S]$ must be a subgraph with no $(k-1)$-cliques (any $(k-1)$-clique in $G[S]$ would form a $k$-clique together with $m$). By induction, there is some $(k-2)$-partite graph $K$ on $|S|$ vertices with $|E(K)| \ge |E(G[S])|$.
Now modify $H$ by replacing $G[S]$ (or $H[S]$, which is the same thing) with the $(k-2)$-partite graph $K$. The result (call it $H'$)
This proves the theorem.