Interval edge-colorings of complete graphs

Hrant Khachatrian, Petros A. Petrosyan

Abstract: An edge-coloring of a graph G<math><mi is="true">G</mi></math> with colors 1,2,…,t<math><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">t</mi></math> is an interval t<math><mi is="true">t</mi></math>-coloring if all colors are used, and the colors of edges incident to each vertex of G<math><mi is="true">G</mi></math> are distinct and form an interval of integers. A graph G<math><mi is="true">G</mi></math> is interval colorable if it has an interval t<math><mi is="true">t</mi></math>-coloring for some positive integer t<math><mi is="true">t</mi></math>. For an interval colorable graph G<math><mi is="true">G</mi></math>, W(G)<math><mi is="true">W</mi><mrow is="true"><mo is="true">(</mo><mi is="true">G</mi><mo is="true">)</mo></mrow></math> denotes the greatest value of t<math><mi is="true">t</mi></math> for which G<math><mi is="true">G</mi></math> has an interval t<math><mi is="true">t</mi></math>-coloring. It is known that the complete graph is interval colorable if and only if the number of its vertices is even. However, the exact value of W(K2n)<math><mi is="true">W</mi><mrow is="true"><mo is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">K</mi></mrow><mrow is="true"><mn is="true">2</mn><mi is="true">n</mi></mrow></msub><mo is="true">)</mo></mrow></math> is known only for n≤4<math><mi is="true">n</mi><mo is="true">≤</mo><mn is="true">4</mn></math>. The second author showed that if n=p2q<math><mi is="true">n</mi><mo is="true">=</mo><mi is="true">p</mi><msup is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mi is="true">q</mi></mrow></msup></math>, where p<math><mi is="true">p</mi></math> is odd and q<math><mi is="true">q</mi></math> is nonnegative, then W(K2n)≥4n−2−p−q<math><mi is="true">W</mi><mrow is="true"><mo is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">K</mi></mrow><mrow is="true"><mn is="true">2</mn><mi is="true">n</mi></mrow></msub><mo is="true">)</mo></mrow><mo is="true">≥</mo><mn is="true">4</mn><mi is="true">n</mi><mo is="true">−</mo><mn is="true">2</mn><mo is="true">−</mo><mi is="true">p</mi><mo is="true">−</mo><mi is="true">q</mi></math>. Later, he conjectured that if n∈N<math><mi is="true">n</mi><mo is="true">∈</mo><mi mathvariant="double-struck" is="true">N</mi></math>, then W(K2n)=4n−2−⌊log2n⌋−‖n2‖<math><mi is="true">W</mi><mrow is="true"><mo is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">K</mi></mrow><mrow is="true"><mn is="true">2</mn><mi is="true">n</mi></mrow></msub><mo is="true">)</mo></mrow><mo is="true">=</mo><mn is="true">4</mn><mi is="true">n</mi><mo is="true">−</mo><mn is="true">2</mn><mo is="true">−</mo><mrow is="true"><mo is="true">⌊</mo><msub is="true"><mrow is="true"><mo is="true">log</mo></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub><mi is="true">n</mi><mo is="true">⌋</mo></mrow><mo is="true">−</mo><mrow is="true"><mo is="true">‖</mo><msub is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub><mo is="true">‖</mo></mrow></math>, where ‖n2‖<math><mrow is="true"><mo is="true">‖</mo><msub is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mn is="true">2</mn></mrow></msub><mo is="true">‖</mo></mrow></math> is the number of 1’s in the binary representation of n<math><mi is="true">n</mi></math>.In this paper we introduce a new technique to construct interval colorings of complete graphs based on their 1-factorizations, which is used to disprove the conjecture, improve lower and upper bounds on W(K2n)<math><mi is="true">W</mi><mrow is="true"><mo is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">K</mi></mrow><mrow is="true"><mn is="true">2</mn><mi is="true">n</mi></mrow></msub><mo is="true">)</mo></mrow></math> and determine its exact values for n≤12<math><mi is="true">n</mi><mo is="true">≤</mo><mn is="true">12</mn></math>.