  An edge-coloring of a graph $G$ with colors $1,2,\ldots,t$ 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 $W(K_{2n})$ is known only for $n \leq 4$. The
second author showed that if $n = p2^q$, where $p$ is odd and $q$ is
nonnegative, then $W(K_{2n}) \geq 4n-2-p-q$. Later, he conjectured that if $n
\in \mathbb{N}$, then $W(K_{2n}) = 4n - 2 - \left\lfloor\log_2{n}\right\rfloor
- \left \| n_2 \right \|$, where $\left \| n_2 \right \|$ 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 $W(K_{2n})$ and determine its
exact values for $n \leq 12$.
