关于大最小度平衡三部图中三角形的最小数量
On the minimum number of triangles in balanced tripartite graphs with large minimum degree
- 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）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究针对三部图最小度约束下三角形最小数量问题，构造反例推翻Bollobás等人提出的$4t^3$上界猜想，并将下界从$t^3$改进至$\frac{12}{5}t^3$。
AI中文摘要：
设$f(n,t)$为每个部分有$n$个顶点且最小度至少为$n+t$的三部图中三角形的最小数量。1975年，Bollobás、Erdős和Szemerédi证明了$f(n,1)=\min\{4,n\}$。他们进一步指出，对于$n\ge5t$，$f(n,t)\ge4t^3$“非常可能”成立。他们还证明了对于所有整数$n\ge t \ge 1$，$f(n,t) \ge t^3$。我们构造的图表明，对于所有整数$t\ge1$和$n\ge3t+2\lceil(1+\sqrt5)t/2\rceil$，\\[ f(n,t)\le(1+\sqrt5)t^3+\left(1+\frac1{\sqrt5}\right)t^2. \\] 这里$1+\sqrt5\approx3.236<4$，且对于每个$t\ge2$，上述上界严格小于$4t^3$，从而反驳了他们提出的界。我们还通过证明对于所有整数$t\ge2$和$n\ge18t^6$，$f(n,t)\ge\frac{12}{5}t^3$，改进了他们的下界$t^3$。
英文摘要：
Let $f(n,t)$ be the minimum number of triangles in a tripartite graph with $n$ vertices in each part and minimum degree at least $n+t$. In 1975, Bollobás, Erdős and Szemerédi proved that $f(n,1)=\min\{4,n\}$. They further remarked that it is ``very likely'' that $f(n,t)\ge4t^3$ for $n\ge5t$. They also proved that $f(n,t) \ge t^3$ for all integers $n \ge t \ge 1$. We construct graphs showing that, for all integers $t\ge1$ and $n\ge3t+2\lceil(1+\sqrt5)t/2\rceil$, \[ f(n,t)\le(1+\sqrt5)t^3+\left(1+\frac1{\sqrt5}\right)t^2. \] Here $1+\sqrt5\approx3.236<4$, and the displayed upper bound is strictly less than $4t^3$ for every $t\ge2$, disproving their proposed bound. We also improve their lower bound $t^3$ by showing that $f(n,t)\ge\frac{12}{5}t^3$ for all integers $t\ge2$ and $n\ge18t^6$.
