اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدTuran numbers and anti-Ramsey numbers for short cycles in complete $3$-partite graphs
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We call a $4$-cycle in $K_{n_{1}, n_{2}, n_{3}}$ multipartite, denoted by $C_{4}^{text{multi}}$, if it contains at least one vertex in each part of $K_{n_{1}, n_{2}, n_{3}}$. The Turan number $text{ex}(K_{n_{1},n_{2},n_{3}}, C_{4}^{text{multi}})$ $bigg($ respectively, $text{ex}(K_{n_{1},n_{2},n_{3}},{C_{3}, C_{4}^{text{multi}}})$ $bigg)$ is the maximum number of edges in a graph $Gsubseteq K_{n_{1},n_{2},n_{3}}$ such that $G$ contains no $C_{4}^{text{multi}}$ $bigg($ respectively, $G$ contains neither $C_{3}$ nor $C_{4}^{text{multi}}$ $bigg)$. We call a $C^{multi}_4$ rainbow if all four edges of it have different colors. The ant-Ramsey number $text{ar}(K_{n_{1},n_{2},n_{3}}, C_{4}^{text{multi}})$ is the maximum number of colors in an edge-colored of $K_{n_{1},n_{2},n_{3}}$ with no rainbow $C_{4}^{text{multi}}$. In this paper, we determine that $text{ex}(K_{n_{1},n_{2},n_{3}}, C_{4}^{text{multi}})=n_{1}n_{2}+2n_{3}$ and $text{ar}(K_{n_{1},n_{2},n_{3}}, C_{4}^{text{multi}})=text{ex}(K_{n_{1},n_{2},n_{3}}, {C_{3}, C_{4}^{text{multi}}})+1=n_{1}n_{2}+n_{3}+1,$ where $n_{1}ge n_{2}ge n_{3}ge 1.$
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