Abstract
We show that the union closed sets conjecture holds for tree convex sets. The union closed sets conjecture says that in every union closed set system, there is an element to appear in at least half of the members of the system. In tree convex set systems, there is a tree on the elements such that each subset induces a subtree. Our proof relies on the well known Helly property of hypertrees and an equivalent graph formulation of the union closed sets conjecture given in (Bruhn, H., Charbit, P. and Telle, J.A.: The graph formulation of the union-closed sets conjecture. Proc. of EuroComb 2013, 73–78 (2013)).
Partially supported by Natural Science Foundation of China (Grant Nos. 61370052 and 61370156).
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The help of anonymous reviewers has improved our presentation greatly.
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Liu, T., Xu, K. (2015). Union Closed Tree Convex Sets. In: Wang, J., Yap, C. (eds) Frontiers in Algorithmics. FAW 2015. Lecture Notes in Computer Science(), vol 9130. Springer, Cham. https://doi.org/10.1007/978-3-319-19647-3_19
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DOI: https://doi.org/10.1007/978-3-319-19647-3_19
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