Topological graph passwords based on the gracefulness of disconnected graphs with circles
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摘要: 图的标号是图论的一个重要分支, 从非连通图入手, 证明了两类非连通图
$G=(C_{4}, \overline{K}_{r})\cup$ $ \bigcup _{t=1}^k K^{t}_{m, n}$ 和$G=(C_{4},\overline{K}_{r})\cup K_{1, n}\cup \bigcup_{t=1}^k T^t$ 都具有优美标号. 且证明方法能够算法化, 为非连通图应用于网络提供了可行的理论保证.Abstract: Graph labeling is an important branch of graph theory. We use disconnected graphs to start our study, and show that disconnected graphs$G= (C_{4},\overline{K}_{r})\cup$ $ \bigcup _{t=1}^k K^{t}_{m, n}$ and$G= (C_{4},\overline{K}_{r})\cup$ $ K_{1, n}\cup \bigcup_{t=1}^k T^t$ allow graceful labelings. Algorithms can be used to provide a theoretical basis for proving the application of disconnected graphs to networks.-
Key words:
- disconnected graphs /
- complete bipartite graphs /
- graceful labeling
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图 1
$ G_{0} $ 、$ G_{1} $ 、$ G_{2} $ 的优美标号Fig. 1 The graceful labellng of
$ G_{0} $ ,$ G_{1} $ and$ G_{2} $ 
图 2
$ G = G_{0}\cup G_{1}\cup G_{2} $ 的一个优美标号Fig. 2 The graceful labeling of the union graph
$ G = G_{0}\cup G_{1}\cup G_{2} $ 
图 3 (a)优美图
$ (C_{4},\overline{K}_{6}) $ ; (b)优美图$ K_{3, 4} $ Fig. 3 (a) The graceful graph
$ (C_{4},\overline{K}_{6}) $ ; (b) the graceful graph$ K_{3, 4} $ 
图 4 优美图
$ (C_{4},\overline{K}_{6})\cup \bigcup _{t = 1}^4 K^{t}_{3,4} $ Fig. 4 The graceful graph
$ (C_{4},\overline{K}_{6})\cup \bigcup _{t = 1}^4 K^{t}_{3,4} $ 
图 5 (a)优美
$ (C_{4},\overline{K}_{8}) $ ; (b)优美$ K_{1, 8} $ ; (c)优美$ T_{15} $ ; (d)优美$ T_{8} $ ; (e) 优美$ T_{8} $ Fig. 5 (a) The graceful
$ (C_{4},\overline{K}_{8}) $ ; (b) the graceful$ K_{1, 8} $ ; (c) the graceful$ T_{15} $ ; (d) the graceful$ T_{8} $ ; (e) the graceful$ T_{8} $ 
图 6 优美图
$ (C_{4},\overline{K}_{8})\cup K_{1, 8}\cup \bigcup_{t = 1}^3 T^t $ Fig. 6 The graceful graph
$ (C_{4},\overline{K}_{8})\cup K_{1, 8}\cup \bigcup_{t = 1}^3 T^t $ 
图 7 (a)优美
$ (C_{4},\overline{K}_{r}) $ ; (b)优美$ T_{10} $ ; (c)优美$ T_{8} $ ; (d)优美$ T_{10} $ ; (e)优美$ T_{10} $ Fig. 7 (a) The graceful
$ (C_{4},\overline{K}_{r}) $ ; (b) the graceful$ T_{10} $ ; (c) the graceful$ T_{8} $ ; (d) the graceful$ T_{10} $ ; (e) the graceful$ T_{10} $ -
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