For any graph $G=(V,E)$ let $\tau(G)$ be the minimum cardinality of a vertex cover of $G$. Let $L(G)$ denote the line graph of $G$.
Given any positive integer $n\in\mathbb{N}$, what is the value of $\tau(L(K_n))$ where $K_n$ denotes the complete graph on $n$ points?
A vertex cover of $L(K_n)$ is just a subset of $E(K_n)$ whose complement is a matching. Since a maximum matching in $K_n$ has $\left\lfloor\frac n2\right\rceil$ edges, $$\tau(L(K_n))=\binom n2-\left\lfloor\frac n2\right\rceil=\left\lceil\frac{n^2}2\right\rceil-n.$$