Asymptotic density of graphs excluding disconnected minors

Abstract

For a graph H, letc∞(H)=limn→∞⁡max⁡|E(G)|n, where the maximum is taken over all graphs G on n vertices not containing H as a minor. Thus c∞(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∞(H) for disconnected graphs H. In particular, we determine c∞(H) whenever H is a union of cycles. Finally, we investigate the behaviour of c∞(sKr) for fixed r, where sKr denotes the union of s disjoint copies of the complete graph on r vertices. Improving on a result of Thomason, we show thatc∞(sKr)=s(r−1)−1fors=ω(log⁡rlog⁡log⁡r), andc∞(sKr)>s(r−1)−1fors=o(log⁡rlog⁡log⁡r).