Origins of Nonlinearity in Davenport-Schinzel Sequences
SIAM Journal on Discrete Mathematics
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In this paper we study labeled–tree analogues of (generalized) Davenport–Schinzel sequences.We say that two sequences a1 ... ak, b1 ... bk of equal length k are isomorphic, if ai = aj iffbi = bj (for all i, j). For example, the sequences 11232, 33141 are isomorphic. We investigate the maximum size of a labeled (rooted) tree with each vertex labeled by one of n labels in such a way that, besides some technical conditions, the sequence of labels along any path (starting from the root) contains no subsequence isomorphic to a fixed “forbidden” sequence u.We study two models of such labeled trees. Each of the models is known to be essentially equivalent also to other models. The labeled paths in a special case of one of our models correspond to classical Davenport–Schinzel sequences.We investigate, in particular, for which sequences u the labeled tree has at most O(n) vertices. In both models, we answer this question for any forbidden sequence u over a two-element alphabet and also for a large class of other sequences u.