On Connected 2-K
n
-Residual Graphs |
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Authors: | Jiangdong Liao Shihui Yang Yong Deng |
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Institution: | 1. College of Mathematics and Computer Sciences, Yangtze Normal University, 408100, Chongqing, Fuling, P.R.China 2. Computer and Information Sciences, Southwest University, 400715, Chongqing, P.R.China 3. College of Science, Chongqing Technology and Business University, 400067, Chongqing, P.R.China
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Abstract: | A graph G is said to be K n -residual if for every point u in G, the graph obtained by removing the closed neighborhood of u from G is isomorphic to K n . We inductively define a multiply-K n -residual graph by saying that G is m-K n -residual if the removal of the closed neighborhood of any vertex of G results in an (m – 1)-K n -residual graphs. Erdös, Harary and Klawe 2] determined the minimum order of the m?K n -residual graphs for all m and n, which are not necessarily connected, the minimum order of connected; K n -residual graphs, all K n -residual extremal graphs. They also stated some conjectures regarding the connected case. In this paper, we determine the minimum order of a connected 2-K n -residual graph and specify the extremal graphs, expect for n = 3. In particular, we determining only one connected 2-K 4-residual graph of minimal order, and show that there is a connected 2-K 6-residual graph non isomorphic to K 8 × K 3 with minimum order. Finally we present and a revised version of the conjecture in 2]. |
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