树与星森林的二分Turán数

中文摘要 AI 辅助

图H的二分Turán数记为ex(m, n; H)，是顶点划分为大小分别为|A|=m和|B|=n的二分图G=(A,B;E)中，不含H作为子图的图的最大边数。针对两类图族研究该问题：对于顶点划分为大小|R|=r≤s=|S|的树T=T(r,s)，当n相对于m、r、s足够大时，证明了(r-1)n ≤ ex(m, n; T(r,s)) ≤ (r-1)n + O(m)，精确确定了主项（星情况r=1时用精确公式求解）；对于星森林F=∪_{i=1}^k S_{d_i}（满足d₁≥…≥d_k），当n足够大时，精确确定ex(m, n; F)=(k-1)n + (d_k - 1)(m - k + 1)，并刻画了唯一的极值图。

英文摘要

The bipartite Turán number of a graph $H$, denoted $\text{ex}(m, n; H)$, is the maximum number of edges in any $H$-free bipartite graph $G = (A, B; E)$ with parts of size $|A| = m$ and $|B| = n$. We study this problem for two families. For a tree $T = T(r, s)$ with parts $R$ and $S$ of sizes $|R| = r \le s = |S|$, we prove \[ (r - 1) n \;\le\; \text{ex}(m, n; T(r, s)) \;\le\; (r - 1) n + C(m,r,s) \] for $n$ sufficiently large compared to $m$, $r$, and $s$, where $C(m,r,s)$ does not depend on $n$, determining the leading-order term exactly (with the star case $r = 1$ solved with an exact formula). For a star forest $F = \bigcup_{i=1}^k S_{d_i}$ with $d_1 \ge \cdots \ge d_k$, we determine the exact value $\text{ex}(m, n; F) = (k - 1) n + (d_k - 1)(m - k + 1)$ for $n$ sufficiently large, and characterize the unique extremal graph.

发表机构

• Harvey Mudd College（哈维穆德学院）

机构由 AI 辅助整理，请以论文原文为准。