Abstract
A directed graph is said to ben-unavoidable if it is contained as a subgraph by every tournament onn vertices. A number of theorems have been proven showing that certain graphs aren-unavoidable, the first being Rédei's results that every tournament has a Hamiltonian path. M. Saks and V. Sós gave more examples in [6] and also a conjecture that states: Every directed claw onn vertices such that the outdegree of the root is at most [n/2] isn-unavoidable. Here a claw is a rooted tree obtained by identifying the roots of a set of directed paths. We give a counterexample to this conjecture and prove the following result:any claw of rootdegree≤n/4 is n-unavoidable.
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Lu, X. On claws belonging to every tournament. Combinatorica 11, 173–179 (1991). https://doi.org/10.1007/BF01206360
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DOI: https://doi.org/10.1007/BF01206360