Fault-tolerant hamiltonian connectivity of the WK-recursive networks

Tung-Yang Ho, Cheng-Kuan Lin, Jimmy J. M. Tan, Lih-Hsing Hsu

Abstract: Many research on the WK-recursive network has been published during the past several years due to its favorite properties. In this paper, we consider the fault-tolerant hamiltonian connectivity of the WK-recursive network. We use K(d,t)<math><mrow is="true"><mi is="true">K</mi><mo stretchy="false" is="true">(</mo><mi is="true">d</mi><mtext is="true">,</mtext><mi is="true">t</mi><mo stretchy="false" is="true">)</mo></mrow></math> to denote the WK-recursive network of level t, each of which basic modules is a d-vertex complete graph, where d>1<math><mrow is="true"><mi is="true">d</mi><mo is="true">&gt;</mo><mn is="true">1</mn></mrow></math> and t⩾1<math><mrow is="true"><mi is="true">t</mi><mo is="true">⩾</mo><mn is="true">1</mn></mrow></math>. The fault-tolerant hamiltonian connectivity Hfκ(G)<math><mrow is="true"><msubsup is="true"><mrow is="true"><mi mathvariant="script" is="true">H</mi></mrow><mrow is="true"><mi is="true">f</mi></mrow><mrow is="true"><mi is="true">κ</mi></mrow></msubsup><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo></mrow></math> is defined to be the maximum integer k such that G is k fault-tolerant hamiltonian connected if G is hamiltonian connected and is undefined otherwise. In this paper, we prove that Hfκ(K(d,t))=d-4<math><mrow is="true"><msubsup is="true"><mrow is="true"><mi mathvariant="script" is="true">H</mi></mrow><mrow is="true"><mi is="true">f</mi></mrow><mrow is="true"><mi is="true">κ</mi></mrow></msubsup><mo stretchy="false" is="true">(</mo><mi is="true">K</mi><mo stretchy="false" is="true">(</mo><mi is="true">d</mi><mtext is="true">,</mtext><mi is="true">t</mi><mo stretchy="false" is="true">)</mo><mo stretchy="false" is="true">)</mo><mo is="true">=</mo><mi is="true">d</mi><mo is="true">-</mo><mn is="true">4</mn></mrow></math> if d⩾4<math><mrow is="true"><mi is="true">d</mi><mo is="true">⩾</mo><mn is="true">4</mn></mrow></math>.