Asymptotic Density of Graphs Excluding Disconnected Minors

Abstract

For a graph H, letwhere 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 union of cycles. Finally, we investigate the behaviour of c ∞ (sK r ) for fixed r, where sK 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 ∞ (sK r ) = s(r − 1) − 1 for s = Ω log r log log r , and c ∞ (sK r ) > s(r − 1) − 1 for s = o log r log log r .

A Lower Bound on the Average Degree Forcing a Minor

“…To formalise this question, let f (H) be the infimum of all real numbers d such that every graph with average degree at least d contains H as a minor. This value has been extensively studied for numerous graphs H, including small complete graphs [6,12,21,28,29], the Petersen graph [11], general complete graphs [2,5,14,15,21,23,30,31], complete bipartite graphs [3,16,17,[17][18][19]24], general dense graphs [25], general sparse graphs [9,26,27], disjoint unions of graphs [4,13,33], and disjoint unions of cycles [8]; see [32] for a survey.…”