Cyclic edge and cyclic vertex connectivity of (4,5,6)-fullerene graphs

Jun Liang, Xinyao Liu, Dingjun Lou, Zan-Bo Zhang, Zixin Qin

Abstract: Cyclic vertex connectivity cκ<math><msub is="true"><mrow is="true"><mi is="true">c</mi></mrow><mrow is="true"><mi is="true">κ</mi></mrow></msub></math> and cyclic edge connectivity cλ<math><msub is="true"><mrow is="true"><mi is="true">c</mi></mrow><mrow is="true"><mi is="true">λ</mi></mrow></msub></math> are two important kinds of conditional connectivity, which reflect the number of vertices or edges that can be removed before the graph is disconnected and at least two components contain a cycle, respectively. They have important applications in various networks such as computer networks or biochemical networks. In addition, a fullerene is a special kind of molecule in chemistry. A classic fullerene graph is a 3-connected cubic planar graph with only pentagonal and hexagonal faces. (4, 5, 6)-fullerene graphs are atypical fullerene graphs which also contain 4-faces. In this paper, we prove that cκ=cλ<math><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">c</mi></mrow><mrow is="true"><mi is="true">κ</mi></mrow></msub><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><msub is="true"><mrow is="true"><mi is="true">c</mi></mrow><mrow is="true"><mi is="true">λ</mi></mrow></msub></mrow></math> for (4,5,6)<math><mrow is="true"><mo is="true">(</mo><mn is="true">4</mn><mo is="true">,</mo><mn is="true">5</mn><mo is="true">,</mo><mn is="true">6</mn><mo is="true">)</mo></mrow></math>-fullerene graphs except for four exceptional graphs with order less than 16. We also give O(ν)<math><mrow is="true"><mi is="true">O</mi><mrow is="true"><mo is="true">(</mo><mi is="true">ν</mi><mo is="true">)</mo></mrow></mrow></math>-algorithms to determine the cyclic vertex connectivity and the cyclic edge connectivity of (4,5,6)<math><mrow is="true"><mo is="true">(</mo><mn is="true">4</mn><mo is="true">,</mo><mn is="true">5</mn><mo is="true">,</mo><mn is="true">6</mn><mo is="true">)</mo></mrow></math>-fullerene graphs.