On the purity of minor-closed classes of graphs
نویسنده
چکیده
Given a graphH with at least one edge, let gap H (n) denote the maximum difference between the numbers of edges in two n-vertex edge-maximal graphs with no minor H . We show that for exactly four connected graphs H (with at least two vertices), the class of graphs with no minor H is pure, that is, gap H (n) = 0 for all n > 1; and for each connected graph H (with at least two vertices) we have the dichotomy that either gap H (n) = O(1) or gap H (n) = Θ(n). Further, if H is 2-connected and does not yield a pure class, then there is a constant c > 0 such that gap H (n) ∼ cn. We also give some partial results when H is not connected or when there are two or more excluded minors.
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