| Abstract: | 一個不含 K_2,2 的二分極圖是一個含有最多邊數且不含有子圖 K_2,2 的二分圖。若此二分圖為 K_m,n 的子圖，則求此二分極圖的邊數的問題也就是出名的Zarankiewicz 問題。在本論文中，我們令f(m,n)為 K_m,n 中不含 K_2,2 的二分極圖的邊數。我們獲得以下的結果：
①f(m,n)≤n/2+√(mn(m-1)+n^2/4)
②若 n≥(m|2)，則 f(m,n)=(m|2)+n。
③若 m≡1,3 (mod 6)，則 K_m 恰可分割成 m(m-1)/6 個邊相異的 K_3 子圖，且f(m,n)=(m|2)。
④若 ((m|2))/3≤n≤(m|2)，則 f(m,n)=[((m|2)+3n)/2] 。
⑤若 m≡1,4 (mod 12)，則 K_m 恰可分割成 m(m-1)/12 個邊相異的 K_4 子圖，且，f(m,n)=2(m|2)/3。
An extremal K_2,2-free bipartite graphs is a bipartite graph which contains the maximum number of edges and does not contain any subgraph K_2,2. If this bipartite graph is a subgraph of K_m,n, then finding the number of edges of the extremal bipartite graph is the well-known Zarankiewicz Problem. In this thesis, we let f(m,n) be the number of edges of the extremal K_2,2-free bipartite graph which is a subgraph of K_m,n. We obtain the following results:
①f(m,n)≤n/2+√(mn(m-1)+n^2/4)
②If ≥(m|2), then f(m,n)=(m|2)+n.
③If m≡1,3 (mod 6), then K_m decompose into (m(m-1))/6 edge-disjoint K_3 subgraphs, and f(m,n)=(m|2).
④If ((m|2))/3≤n≤(m|2), then f(m,n)=[((m|2)+3n)/2] .
⑤If m≡1,4 (mod 12), then K_m decompose into (m(m-1))/12 edge-disjoint K_4 subgraphs, and f(m,n)=2(m|2)/3. |
