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- W3123485365 abstract "Let $U_5$ be the tournament with vertices $v_1$, ..., $v_5$ such that $v_2 rightarrow v_1$, and $v_i rightarrow v_j$ if $j-i equiv 1$, $2 pmod{5}$ and ${i,j} neq {1,2}$. In this paper we describe the tournaments which do not have $U_5$ as a subtournament. Specifically, we show that if a tournament $G$ is prime---that is, if there is no subset $X subseteq V(G)$, $1 < |X| < |V(G)|$, such that for all $v in V(G) backslash X$, either $v rightarrow x$ for all $x in X$ or $x rightarrow v$ for all $x in X$---then $G$ is $U_5$-free if and only if either $G$ is a specific tournament $T_n$ or $V(G)$ can be partitioned into sets $X$, $Y$, $Z$ such that $X cup Y$, $Y cup Z$, and $Z cup X$ are transitive. From the prime $U_5$-free tournaments we can construct all the $U_5$-free tournaments. We use the theorem to show that every $U_5$-free tournament with $n$ vertices has a transitive subtournament with at least $n^{log_3 2}$ vertices, and that this bound is tight." @default.
- W3123485365 created "2021-02-01" @default.
- W3123485365 creator A5054884992 @default.
- W3123485365 date "2014-03-13" @default.
- W3123485365 modified "2023-10-17" @default.
- W3123485365 title "Structure Theorem forU5-free Tournaments" @default.
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- W3123485365 doi "https://doi.org/10.1002/jgt.21789" @default.
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