Fault-Hamiltonicity of Hypercube-Like Interconnection Networks

Abstract

We call a graph G to be f-fault Hamiltonian (resp. f-fault Hamiltonian-connected,) if there exists a Hamiltonian cycle (resp. if each pair of vertices are joined by a Hamiltonian path) in G /spl bsol/ F for any set F of faulty elements with |F| /spl les/ f. In this paper, 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/. Provided each G/sub i/ is f-fault Hamiltonian-connected and f+1-fault Hamiltonian, 0 /spl les/ i /spl les/ 3, we show that G/sub 0/ /spl oplus/ G/sub 1/ is f+1-fault Hamiltonian-connected for any f /spl ges/ 2 and f+2-fault Hamiltonian for any f /spl ges/ 1, and that for any f /spl ges/ 0, H/sub 0/ /spl oplus/ H/sub 1/ is f+2-fault Hamiltonian-connected and f+3-fault Hamiltonian, where H/sub 0/ = G/sub 0/ /spl oplus/ G/sub 1/ and H/sub 1/ = G/sub 2/ /spl oplus/ G/sub 3/. 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/. Applying our main results to a subclass of hypercube-like interconnection networks, called restricted HL-graphs, which include twisted-cubes, crossed cubes, multiply twisted cubes, Mobius cubes, Mcubes, and generalized twisted cubes, we show that every restricted HL-graph of degree m( /spl ges/ 3) is m - 3-fault Hamiltonian-connected and m - 2-fault Hamiltonian.