Abstract
A many-to-many k-disjoint path cover (k-DPC) of a graph G is a set of k disjoint paths joining k sources and k sinks in which each vertex of G is covered by a path. It is called a paired many-to-many disjoint path cover when each source should be joined to a specific sink, and it is called an unpaired many-to-many disjoint path cover when each source can be joined to an arbitrary sink. In this paper, we discuss about paired and unpaired many-to-many disjoint path covers including their relationships, application to strong Hamiltonicity, and necessary conditions. And then, we give a construction scheme for paired many-to-many disjoint path covers in the graph H0 oplus H1 obtained from connecting two graphs H0 and H1 with |V(H0)| = |V(H1)| by |V(H1)| pairwise nonadjacent edges joining vertices in H0 and vertices in H1, where H0 = G0 oplus G1 and H1 = G2 oplus G3 for some graphs Gj. Using the construction, we show that every m-dimensional restricted HL-graph and recursive circulant G(2m, 4) with f or less faulty elements have a paired k-DPC for any f and k ges 2 with f + 2k les m.
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