The cubic graphs with finite cyclic vertex connectivity larger than girth

Jun Liang, Dingjun Lou, Zan-Bo Zhang

Abstract: Cyclic (vertex and edge) connectivity is an important concept in graphs. While cyclic edge connectivity (cλ<math><mrow is="true"><mi is="true">c</mi><mi is="true">λ</mi></mrow></math>) has been studied for many years, the study at cyclic vertex connectivity (cκ<math><mrow is="true"><mi is="true">c</mi><mi is="true">κ</mi></mrow></math>) is still at the initial stage. And cκ<math><mrow is="true"><mi is="true">c</mi><mi is="true">κ</mi></mrow></math> seems to be more complicated than cλ<math><mrow is="true"><mi is="true">c</mi><mi is="true">λ</mi></mrow></math>. We have got a sufficient condition that ν(G)≥2g(k−1)<math><mrow is="true"><mi is="true">ν</mi><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow><mo linebreak="goodbreak" linebreakstyle="after" is="true">≥</mo><mn is="true">2</mn><mi is="true">g</mi><mrow is="true"><mo is="true">(</mo><mi is="true">k</mi><mo is="true">−</mo><mn is="true">1</mn><mo is="true">)</mo></mrow></mrow></math> for cκ≠∞<math><mrow is="true"><mi is="true">c</mi><mi is="true">κ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">≠</mo><mi is="true">∞</mi></mrow></math>. On the other hand, if ν(G)<2g(k−1)<math><mrow is="true"><mi is="true">ν</mi><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow><mo linebreak="goodbreak" linebreakstyle="after" is="true">&lt;</mo><mn is="true">2</mn><mi is="true">g</mi><mrow is="true"><mo is="true">(</mo><mi is="true">k</mi><mo is="true">−</mo><mn is="true">1</mn><mo is="true">)</mo></mrow></mrow></math>, then we have cκ=∞<math><mrow is="true"><mi is="true">c</mi><mi is="true">κ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mi is="true">∞</mi></mrow></math>, or cκ≤(k−2)g<math><mrow is="true"><mi is="true">c</mi><mi is="true">κ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">≤</mo><mrow is="true"><mo is="true">(</mo><mi is="true">k</mi><mo is="true">−</mo><mn is="true">2</mn><mo is="true">)</mo></mrow><mi is="true">g</mi></mrow></math>, or (k−2)g<cκ<∞<math><mrow is="true"><mrow is="true"><mo is="true">(</mo><mi is="true">k</mi><mo is="true">−</mo><mn is="true">2</mn><mo is="true">)</mo></mrow><mi is="true">g</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&lt;</mo><mi is="true">c</mi><mi is="true">κ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&lt;</mo><mi is="true">∞</mi></mrow></math>. So characterizing all the k<math><mi is="true">k</mi></math>-regular graphs with (k−2)g<cκ<∞<math><mrow is="true"><mrow is="true"><mo is="true">(</mo><mi is="true">k</mi><mo is="true">−</mo><mn is="true">2</mn><mo is="true">)</mo></mrow><mi is="true">g</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&lt;</mo><mi is="true">c</mi><mi is="true">κ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&lt;</mo><mi is="true">∞</mi></mrow></math> is helpful to design an efficient algorithm for cκ<math><mrow is="true"><mi is="true">c</mi><mi is="true">κ</mi></mrow></math>. Hence, we characterize all 38 cubic graphs with g<cκ<∞<math><mrow is="true"><mi is="true">g</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&lt;</mo><mi is="true">c</mi><mi is="true">κ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">&lt;</mo><mi is="true">∞</mi></mrow></math> and prove that cκ=g+1<math><mrow is="true"><mi is="true">c</mi><mi is="true">κ</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><mi is="true">g</mi><mo linebreak="goodbreak" linebreakstyle="after" is="true">+</mo><mn is="true">1</mn></mrow></math>.