| Summary: | 博士 === 國立交通大學 === 資訊科學與工程研究所 === 99 === The Menger Theorem (Menger, 1972) show that there are k-disjoint paths between any two distinct vertices of G if and only if G is k-connected. A graph G is k*-connected if there exists a k-disjoint paths between any two distinct vertices which contains all the vertices of G. The
spanning connectivity of G, k*(G), is defined as
the largest integer k such that G is w*-connected for 0 < e w < k+1 if G is a 1*-connected graph.
We study the problem of spanning
connectivity properties on some interconnection networks and graphs.
Cycle embedding and path embedding are perhaps the most important outstanding materials in graph theory and have been defying solutions for more than a century. A hamiltonian cycle C of graph G is described as < u1, u2, ..., un(G), u1 > to emphasize the order of vertices in C. Thus, u1 is the starting vertex and ui is the i-th vertex in C. Two hamiltonian cycles C1 = < u1, u2, ..., un(G), u1 > and C2 = < v1, v2, ..., vn(G), v1 > of G
are independent if u1 = v1 and ui is not equal to vi for every i in { 2, ..., n(G) }. A set of hamiltonian cycles { C1, C2, ..., Ck } of G are mutually independent if its elements are pairwise independent. The mutually independent hamiltonianicity IHC(G) of graph G the maximum integer k such that for any vertex u of G there exist k mutually independent hamiltonian cycles of G starting at u. We study this problem on some interconnection networks and graphs.
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