Title | ||
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Minimum Degree Conditions and Optimal Graphs for Completely Independent Spanning Trees. |
Abstract | ||
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Completely independent spanning trees (T_1,T_2,ldots ,T_k) in a graph G are spanning trees in G such that for any pair of distinct vertices u and v, the k paths in the spanning trees between u and v mutually have no common edge and no common vertex except for u and v. The concept finds applications in fault-tolerant communication problems in a network. Recently, it was shown that Dirac’s condition for a graph to be hamiltonian is also a sufficient condition for a graph to have two completely independent spanning trees. In this paper, we generalize this result to three or more completely independent spanning trees. Namely, we show that for any graph G with (n ge 7) vertices, if the minimum degree of a vertex in G is at least (n-k), where (3 le k le frac{n}{2}), then there are (lfloor frac{n}{k} rfloor ) completely independent spanning trees in G. Besides, we improve the lower bound of (frac{n}{2}) on the Dirac’s condition for completely independent spanning trees to (frac{n-1}{2}) except for some specific graph. Our results are theoretical ones, since these minimum degree conditions can be applied only to a very dense graph. We then present constructions of symmetric regular graphs which include optimal graphs with respect to the number of completely independent spanning trees. |
Year | Venue | Field |
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2015 | IWOCA | Combinatorics,Minimum degree spanning tree,Vertex (geometry),Upper and lower bounds,Degree (graph theory),Spanning tree,Mathematics,Maximal independent set,Minimum spanning tree,Dense graph |
DocType | Citations | PageRank |
Conference | 7 | 0.54 |
References | Authors | |
9 | 1 |
Name | Order | Citations | PageRank |
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Toru Hasunuma | 1 | 142 | 16.00 |