A disjoint path cover (DPC for short) of a graph is a set of disjoint paths that cover all the vertices of the graph. A paired many-to-many k-DPC is a DPC composed of k paths between k sources and k sinks, such that each source is joined to a designated sink. We show that recursive circulant G(2m,4) with at most f faulty vertices and/or edges being removed has a paired many-to-many k-DPC joining k arbitrary sources and sinks for any f and k ≥ 2, subject to f+2k ≤ m+1, where m ≥ 5. The bound m+1 on f+2k is the best possible.

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 H0oplus H1obtained from connecting two graphs H0and H1with |V(H0)| = |V(H1)| by |V(H1)| pairwise nonadjacent edges joining vertices in H0and vertices in H1, where H0= G0oplus G1and H1= G2oplus G3for 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.

A paired many-to-many k-disjoint path cover (k-DPC) of a graph G is a set of k disjoint paths joining k distinct source-sink pairs in which each vertex of G is covered by a path. This paper is concerned with paired many-to-many disjoint path coverability of hypercube-like interconnection networks, called restricted HL-graphs. The class includes twisted cubes, crossed cubes, multiply twisted cubes, Möbius cubes, Mcubes, and generalized twisted cubes. We show that every restricted HL-graph of degree m with f or less faulty elements has a paired many-to-many k-DPC for any f and k ≥ 2 with f + 2k ≤ m. The result improves the known bound of f + 2k ≤ m − 1 by one.

A many-to-many k-disjoint path cover (k-DPC) of a graph G is a set of k disjoint paths joining k distinct source-sink pairs in which each vertex of G is covered by a path. We deal with the graph G/sub 0/ /spl oplus/ G/sub 1/ obtained from connecting two graphs G/sub 0/ and G/sub 1/ with n vertices each by n pairwise nonadjacent edges joining vertices in G/sub 0/ and vertices in G/sub 1/. Many interconnection networks such as hypercube-like interconnection networks can be represented in the form of G/sub 0/ /spl oplus/ G/sub 1/ connecting two lower dimensional networks G/sub 0/ and G/sub 1/. In the presence of faulty vertices and/or edges, we investigate many-to-many disjoint path coverability of G/sub 0/ /spl oplus/ G/sub 1/ and (G/sub 0/ /spl oplus/ G/sub 1/) /spl oplus/ (G/sub 2/ /spl oplus/ G/sub 3/ ), provided some conditions on the Hamiltonicity and disjoint path coverability of each graph G/sub i/ are satisfied, 0 /spl les/ i /spl les/ 3. We apply our main results to recursive circulant G(2/sup m/, 4) and a subclass of hypercube-like interconnection networks, called restricted HL-graphs. The subclasses includes twisted cubes, crossed cubes, multiply twisted cubes, Mobius cubes, Mcubes, and generalized twisted cubes. We show that all these networks of degree m with f or less faulty elements have a many-to-many k-DPC joining any k distinct source-sink pairs for any k /spl ges/ 1 and f /spl ges/ 0 such that f+2k /spl les/ m - 1.