| Summary: | 碩士 === 淡江大學 === 中等學校教師在職進修數學教學碩士學位班 === 101 === Let G be a graph. If there is no 4-cycle contained in G and connecting any non
adjacent vertices of G will obtain a 4-cycle, then we call G is a C4 saturated graph.
Let sat(n, C4) and ex(n, C4) be the minimum and maximum number of edges of all C4
saturated graphs with n vertices, respectively.
In this thesis, we obtain a construction of C4 saturated graph with n points, and
give another construction of C4 saturated graph with minimum edge. For n <= 11, we give a C4 saturated graph with n vertices and q edges, for each q between sat(n, C4)
and ex(n, C4). After that, we use Maple to check whether the graph is a C4 saturated graph by using the adjacency matrix of a graph and the properties of C4 saturated graphs.
We define a C4 saturated graph in a complete multipartite graph Kn (m) and obtain the following results:
1. sat(Kn,m, C4) <= m + n - 1; and
2. sat(Kn(m), C4) <= mn – 1 + ┌(n-2)/2┐*m, where sat(K, C4) = min {|E(G)|: G is a C4 saturated graph in the graph K} and ┌x┐ is the smallest integer greater than x .
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