Bounds on the convex label number of trees


A convex labelling of a tree is an assignment of distinct non-negative integer labels to vertices such that wheneverx, y andz are the labels of vertices on a path of length 2 theny≦(x+z)/2. In addition if the tree is rooted, a convex labelling must assign 0 to the root. The convex label number of a treeT is the smallest integerm such thatT has a convex labelling with no label greater thanm. We prove that every rooted tree (and hence every tree) withn vertices has convex label number less than 4n. We also exhibitn-vertex trees with convex label number 4n/3+o(n), andn-vertex rooted trees with convex label number 2n +o(n).

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The research by M. B. and A. W. was partly supported by NSF grant MCS—8311422.

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Bern, M., Wong, A. & Klawe, M. Bounds on the convex label number of trees. Combinatorica 7, 221–230 (1987).

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