Asymptotic Density of Graphs Excluding Disconnected Minors

For a graph $H$, let $$c_{\mathrm{\infty }}(H)=\underset{n\to \mathrm{\infty }}{\lim }\max \frac{\mathrm{\mid }E(G)\mathrm{\mid }}{n},$$ where the maximum is taken over all graphs $G$ on $n$ vertices not containing $H$ as a minor. Thus $c_{\mathrm{\infty }}(H)$ is the asymptotic maximum density of graphs not containing $H$ as a minor. Employing a structural lemma due to Eppstein, we prove new upper bounds on $c_{\mathrm{\infty }}(H)$ for disconnected graphs $H$. In particular, we determine $c_{\mathrm{\infty }}(H)$ whenever $H$ is union of cycles. Finally, we investigate the behaviour of $c_{\mathrm{\infty }}(s{K}_r)$ for fixed $r$, where $s{K}_r$ denotes the union of $s$ disjoint copies of the complete graph on $r$ vertices. Improving on a result of Thomason, we show that $$c_{\mathrm{\infty }}(s{K}_r)=s(r-1)-1\mathrm{f}\mathrm{o}\mathrm{r}s=\mathrm{\Omega }\left(\frac{\log r}{\log \log r}\right),$$ and $$c_{\mathrm{\infty }}(s{K}_r)>s(r-1)-1\mathrm{f}\mathrm{o}\mathrm{r}s=o\left(\frac{\log r}{\log \log r}\right)\mathrm{.}$$