论文标题
彩虹循环与彩虹路径
Rainbow cycles vs. rainbow paths
论文作者
论文摘要
如果$ f $的每个边缘都有独特的颜色，则边缘色图$ f $是{\ it rainbow}。 {\ it rainbowturán编号} $ \ mathrm {ex}^*（n，f）图$ f $的$是适当边缘色的$ n $ vertex图中的最大边数，没有$ f $的彩虹副本。 Keevash，Mubayi，Sudakov和Verstraëte对RainbowTurán数字进行了研究。约翰逊（Johnson）和罗姆巴赫（Rombach）介绍了以下广义图turán问题：对于固定图$ h $和$ f $，让$ \ mathrm {ex}^*（n，h，h，f）$表示$ n $ n $ n $ n $ vertex的$ h $的最大彩虹副本，$ n $ n $ - vertex copplytex coplytex coplytex offex offex offex offex offex offex offex offex offex coply no $ f $ f $ f $ f $ f $ f $ f $。 在本文中，我们调查了$ \ mathrm {ex}^*（n，c_ \ ell，p_ \ ell）$的情况，并给出一般的上限以及$ \ ell = 3,4,5 $的确切结果。一路上，我们在$ \ mathrm {ex}^*（n，p_5）$上建立了一个新的最佳上限。我们的主要动机源于试图改善$ \ mathrm {ex}^*（n，p_ \ ell）$的界限，这是最近几个手稿的主题。
An edge-colored graph $F$ is {\it rainbow} if each edge of $F$ has a unique color. The {\it rainbow Turán number} $\mathrm{ex}^*(n,F)$ of a graph $F$ is the maximum possible number of edges in a properly edge-colored $n$-vertex graph with no rainbow copy of $F$. The study of rainbow Turán numbers was introduced by Keevash, Mubayi, Sudakov, and Verstraëte. Johnson and Rombach introduced the following rainbow-version of generalized Turán problems: for fixed graphs $H$ and $F$, let $\mathrm{ex}^*(n,H,F)$ denote the maximum number of rainbow copies of $H$ in an $n$-vertex properly edge-colored graph with no rainbow copy of $F$. In this paper we investigate the case $\mathrm{ex}^*(n,C_\ell,P_\ell)$ and give a general upper bound as well as exact results for $\ell = 3,4,5$. Along the way we establish a new best upper bound on $\mathrm{ex}^*(n,P_5)$. Our main motivation comes from an attempt to improve bounds on $\mathrm{ex}^*(n,P_\ell)$, which has been the subject of several recent manuscripts.
