Extremal problems on consecutive L(2, 1)-labelling

Changhong Lu, Lei Chen, Mingqing Zhai

Abstract: For a given graph G of order n, a k-L(2,1)<math><mi is="true">L</mi><mo stretchy="false" is="true">(</mo><mn is="true">2</mn><mo is="true">,</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></math>-labelling is defined as a function f:V(G)→{0,1,2,…,k}<math><mi is="true">f</mi><mo is="true">:</mo><mi is="true">V</mi><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo><mo is="true">→</mo><mo stretchy="false" is="true">{</mo><mn is="true">0</mn><mo is="true">,</mo><mn is="true">1</mn><mo is="true">,</mo><mn is="true">2</mn><mo is="true">,</mo><mo is="true">…</mo><mo is="true">,</mo><mi is="true">k</mi><mo stretchy="false" is="true">}</mo></math> such that |f(u)-f(v)|⩾2<math><mo is="true">|</mo><mi is="true">f</mi><mo stretchy="false" is="true">(</mo><mi is="true">u</mi><mo stretchy="false" is="true">)</mo><mo is="true">-</mo><mi is="true">f</mi><mo stretchy="false" is="true">(</mo><mi is="true">v</mi><mo stretchy="false" is="true">)</mo><mo is="true">|</mo><mo is="true">⩾</mo><mn is="true">2</mn></math> when dG(u,v)=1<math><msub is="true"><mrow is="true"><mi is="true">d</mi></mrow><mrow is="true"><mi is="true">G</mi></mrow></msub><mo stretchy="false" is="true">(</mo><mi is="true">u</mi><mo is="true">,</mo><mi is="true">v</mi><mo stretchy="false" is="true">)</mo><mo is="true">=</mo><mn is="true">1</mn></math> and |f(u)-f(v)|⩾1<math><mo is="true">|</mo><mi is="true">f</mi><mo stretchy="false" is="true">(</mo><mi is="true">u</mi><mo stretchy="false" is="true">)</mo><mo is="true">-</mo><mi is="true">f</mi><mo stretchy="false" is="true">(</mo><mi is="true">v</mi><mo stretchy="false" is="true">)</mo><mo is="true">|</mo><mo is="true">⩾</mo><mn is="true">1</mn></math> when dG(u,v)=2<math><msub is="true"><mrow is="true"><mi is="true">d</mi></mrow><mrow is="true"><mi is="true">G</mi></mrow></msub><mo stretchy="false" is="true">(</mo><mi is="true">u</mi><mo is="true">,</mo><mi is="true">v</mi><mo stretchy="false" is="true">)</mo><mo is="true">=</mo><mn is="true">2</mn></math>. The L(2,1)<math><mi is="true">L</mi><mo stretchy="false" is="true">(</mo><mn is="true">2</mn><mo is="true">,</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></math>-labelling number of G, denoted by λ(G)<math><mi is="true">λ</mi><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo></math>, is the smallest number k such that G has a k-L(2,1)<math><mi is="true">L</mi><mo stretchy="false" is="true">(</mo><mn is="true">2</mn><mo is="true">,</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></math>-labelling. The consecutive L(2,1)<math><mi is="true">L</mi><mo stretchy="false" is="true">(</mo><mn is="true">2</mn><mo is="true">,</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></math>-labelling is a variation of L(2,1)<math><mi is="true">L</mi><mo stretchy="false" is="true">(</mo><mn is="true">2</mn><mo is="true">,</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></math>-labelling under the condition that the labelling f is an onto function. The consecutive L(2,1)<math><mi is="true">L</mi><mo stretchy="false" is="true">(</mo><mn is="true">2</mn><mo is="true">,</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></math>-labelling number of G is denoted by λ¯(G)<math><mover accent="true" is="true"><mrow is="true"><mi is="true">λ</mi></mrow><mrow is="true"><mo stretchy="true" is="true">¯</mo></mrow></mover><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo></math>. Obviously, λ(G)⩽λ¯(G)⩽|V(G)|-1<math><mi is="true">λ</mi><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo><mo is="true">⩽</mo><mover accent="true" is="true"><mrow is="true"><mi is="true">λ</mi></mrow><mrow is="true"><mo stretchy="true" is="true">¯</mo></mrow></mover><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo><mo is="true">⩽</mo><mo is="true">|</mo><mi is="true">V</mi><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo><mo is="true">|</mo><mo is="true">-</mo><mn is="true">1</mn></math> if G admits a consecutive L(2,1)<math><mi is="true">L</mi><mo stretchy="false" is="true">(</mo><mn is="true">2</mn><mo is="true">,</mo><mn is="true">1</mn><mo stretchy="false" is="true">)</mo></math>-labelling. In this paper, we investigate the graphs with λ¯(G)=|V(G)|-1<math><mover accent="true" is="true"><mrow is="true"><mi is="true">λ</mi></mrow><mrow is="true"><mo stretchy="true" is="true">¯</mo></mrow></mover><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo><mo is="true">=</mo><mo is="true">|</mo><mi is="true">V</mi><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo><mo is="true">|</mo><mo is="true">-</mo><mn is="true">1</mn></math> and the graphs with λ¯(G)=λ(G)<math><mover accent="true" is="true"><mrow is="true"><mi is="true">λ</mi></mrow><mrow is="true"><mo stretchy="true" is="true">¯</mo></mrow></mover><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo><mo is="true">=</mo><mi is="true">λ</mi><mo stretchy="false" is="true">(</mo><mi is="true">G</mi><mo stretchy="false" is="true">)</mo></math>, in terms of their sizes, diameters and the number of components.