本論文探討著名的Vizing 猜想：任意兩個圖的卡氏積圖其控制數大於或等於兩個圖的控制數的乘積。我們證明了如果圖G 有k 個完全子圖G_1，G_2....G_k使得gamma_G(UG_i i = 1 to k ) 等於k，對於任意圖H，兩個圖G 和H 的卡氏積圖的控制數一定會大於或等於k 個H 的控制數。

For a graph G, (G) is the domination number of G. Vizing [2] conjectured that gamma(G Box H) >= gamma(G)gamma(H) for any graph G and H, where G Box H is the Cartesian product of graphs G and H. Clark and Suen [1] proved that gamma(G Box H) >= gamma(G)gamma(H) for any graphs G and H. Barcalkin and German [5] proved that Vizing's conjecture holds for some speci c family of graphs. We combine both of their approaches and prove that if G has k disjoint complete subgraphs G1;G2; : : :Gk and gamma_G(UG_i i = 1 to k ) = k, then gamma(G Box H) >= k gamma(H).

[Introduction page 1.]
[Basic notation page 2.]
[History of Vizing conjecture page4.]
[Known Results page7.]
[BG graphs page7.]
[Weaker version of vizing conjecture page8.]
[Results of this thesis page9.]

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(1979), 5-8.

B. Bre sar and D. F. Rall. Fair reception and Vizings conjecture. J.
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