Interval edge-colorings of complete graphs

An edge-coloring of a graph G with colors 1, 2, . . . , t is an interval t-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 = p2, 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 − blog2 nc − ‖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.