| Abstract: | 一個C4與C6飽和二部圖是一個不含有C4與C6的二部圖，且加上任意不相鄰兩點的邊後會形成C4或C6。我們要探討的是一個C4與C6飽和二部圖的邊數為多少。在本論文中，我們令z(m, n)為二部圖Km,n的子圖中為C4與C6飽和圖的最多邊數，s(m, n)為最少邊數。我們獲得以下的結果：
1.s(m ,n)=m+n-1
2.∀n≥3，z(3, n)=n+2、∀n≥4，z(4, n)=n+4，∀n≥6，z(5, n)=n+6
3.若n≥⌊m^2/4⌋，則z(m,n)= ⌊m^2/4⌋+n
4.設 n≥⌊m^2/4⌋，若s(m ,n)≤ x ≤z(m ,n)，則在 K_(m,n) 中存在一個
C4與C6飽和圖其邊數為x。
A C4, C6-saturated bipartite graph is a bipartite graph if it contains no 4-cycle and 6-cycle, but joining any nonadjacent vertices produces a graph that does contain a 4-cycle or 6-cycle. We will find the number of edges of a C4, C6-saturated bipartite graph. We let z(m, n) be the maximum number of edges of a C4,C6-saturated bipartite graph in Km,n, and s(m, n) be the minimum number of edges of a C4, C6-saturated bipartite graph in Km,n.
In this thesis, we obtains the following results：
1.s(m,n)=m+n-1
2.∀n ≥ 3, z(3, n) = n+2, ∀n≥4, z(4, n) = n+4, ∀n≥6, z(5, n)= n+6.
3.If n≥⌊m^2/4⌋,then z(m,n)= ⌊m^2/4⌋+n
4.Let n≥⌊m^2/4⌋. If s(m,n)≤ x ≤z(m,n), then there exists a subgraph
of K_(m,n) which is a C4, C6-saturated graph, and the number of edges is x. |
