On Some Properties of Point-Covering Number of a Graph

Question

Let $G$ be a graph with (point)-connectivity $\kappa$, line-connectivity $\lambda$, minimum degree $\delta$, and point-covering number $\alpha_0$. Is it true that if $G$ is a complete $n$-partite graph, then the parameters mentioned above are all equal?

Answer 1

Yes. By definition $\kappa(K_n)=n-1$. It's easy to see that $\delta(K_n)=n-1$, since each vertex is adjacent to the $n-1$ others. One standard result by Whitney is that $\kappa(G) \le \lambda(G) \le \delta(G)$, so the earlier values give us that $\lambda(K_n)=n-1$. A set is a vertex cover iff its complement is an independent set. $K_n$ has a maximal independent set of size 1, so a minimal vertex cover will have size $n-1$.