Interval edge-colorings of complete graphs

Abstract

An edge-coloring of a graph G with colors 1,2,…,t is an intervalt-coloring if all colors are used, and the colors of edges incident to each vertex of G are distinct and form an interval of integers. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. For an interval colorable graph G, W(G) denotes the greatest value of t for which G has an interval t-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) is known only for n≤4. The second author showed that if n=p2q, where p is odd and q is nonnegative, then W(K2n)≥4n−2−p−q. Later, he conjectured that if n∈N, then W(K2n)=4n−2−⌊log2n⌋−‖n2‖, where ‖n2‖ is the number of 1’s in the binary representation of n.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) and determine its exact values for n≤12.