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Trees with three leaves are ($$n+1$$)-unavoidable. (English) Zbl 1043.05057
Summary: We prove that every tree of order $$n \geqslant 5$$ with three leaves is ($$n+1$$)-unavoidable. More precisely, we prove that every tree $$A$$ with three leaves of order $$n$$ is contained in every tournament $$T$$ of order $$n+1$$ except if $$(T;A)$$ is $$(R_5;S_3^+)$$ or its dual, where $$R_5$$ is the regular tournament on five vertices and $$S_3^+$$ is the outstar of degree three, i.e. the tree consisting of a root dominating three leaves. We then deduce that Sumner’s conjecture is true for trees with four leaves, i.e. every tree of order $$n$$ with four leaves is $$(2n-2)$$-unavoidable.

MSC:
 05C20 Directed graphs (digraphs), tournaments 05C05 Trees
Keywords:
Tournament; Unavoidable; Tree
Full Text:
References:
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