University of Birmingham > Talks@bham > Combinatorics and Probability Seminar > Trees in Tournaments

Trees in Tournaments

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Given an n-vertex oriented tree T, what is the smallest size a tournament G must be, in order to guarantee G contains a copy of T? A strengthening of Sumner’s conjecture poses that it is enough for G to have (n+k-1) vertices, where k is the number of leaves of T. Recently, Dross and Havet used a method of median orders to prove that this is true for arborescences—i.e. trees with edges oriented outwards from a specified root vertex. We show that median orders can make further progress towards (n+k-1), by proving that there exists a constant C such that |G|=(n+Ck) is enough, as well as confirming a separate conjecture that |G|=(n+k-2) is enough, provided we allow n to grow large with k fixed. In this talk we shall discuss these results and further progress that could be made.

Joint work with Richard Montgomery

This talk is part of the Combinatorics and Probability Seminar series.

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