关于任意两部分之间无4-环的三部图中三角形最大数目
On the maximum number of triangles in tripartite graphs with no $4$-cycles between any two parts
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中文总结 AI 辅助
本文研究三部图中任意两部分间无4-环时的最大三角形数，利用新的平面多项式构造，将下界从$(1-o(1))k^{5/3}$改进至$(1-o(1))k^{17/10}$。
中文摘要 AI 辅助
设$G$为一个三部图，每个部分有$k$个顶点，且任意两部分所诱导的二部图不含长度为四的环。Fischer和Matoušek [J. Combin. Theory Ser. A, 2001] 询问此类图中三角形数目的最大值。他们得到了下界$(1-o(1))k^{3/2}$和上界$k^{7/4}+O(k^{3/2})$。Coulter、Matthews和Timmons [J. Combin. Theory Ser. B, 2018] 后来利用有限域上的平面多项式构造了此类图，并将下界改进为$(1-o(1))k^{5/3}$。在本文中，我们使用一组新的平面多项式，进一步将下界改进为$(1-o(1))k^{17/10}$。
英文摘要
Let $G$ be a $3$-partite graph with $k$ vertices in each part such that the bipartite graph induced by any two parts contains no cycle of length four. Fischer and Matoušek [J. Combin. Theory Ser. A, 2001] asked for the maximum number of triangles in such a graph. They obtained the lower bound $(1-o(1))k^{3/2}$ and the upper bound $k^{7/4}+O(k^{3/2})$. Coulter, Matthews and Timmons [J. Combin. Theory Ser. B, 2018] later constructed such graphs using planar polynomials over finite fields and improved the lower bound to $(1-o(1))k^{5/3}$. In this note, we use a new triple of planar polynomials and further improve the lower bound to $(1-o(1))k^{17/10}$.
发表机构
- School of Computer Science and Technology, Dongguan University of Technology（东莞理工学院计算机科学与技术学院）
- School of Mathematical Sciences, Zhejiang Normal University（浙江师范大学数学科学学院）
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