  A \emph{proper $t$-edge-coloring} of a graph $G$ is a mapping $\alpha:
E(G)\rightarrow \{1,\ldots,t\}$ such that all colors are used, and
$\alpha(e)\neq \alpha(e^{\prime})$ for every pair of adjacent edges
$e,e^{\prime}\in E(G)$. If $\alpha $ is a proper edge-coloring of a graph $G$
and $v\in V(G)$, then \emph{the spectrum of a vertex $v$}, denoted by
$S\left(v,\alpha \right)$, is the set of all colors appearing on edges incident
to $v$. \emph{The deficiency of $\alpha$ at vertex $v\in V(G)$}, denoted by
$def(v,\alpha)$, is the minimum number of integers which must be added to
$S\left(v,\alpha \right)$ to form an interval, and \emph{the deficiency
$def\left(G,\alpha\right)$ of a proper edge-coloring $\alpha$ of $G$} is
defined as the sum $\sum_{v\in V(G)}def(v,\alpha)$. \emph{The deficiency of a
graph $G$}, denoted by $def(G)$, is defined as follows:
$def(G)=\min_{\alpha}def\left(G,\alpha\right)$, where minimum is taken over all
possible proper edge-colorings of $G$. For a graph $G$, the smallest and the
largest values of $t$ for which it has a proper $t$-edge-coloring $\alpha$ with
deficiency $def(G,\alpha)=def(G)$ are denoted by $w_{def}(G)$ and $W_{def}(G)$,
respectively. In this paper, we obtain some bounds on $w_{def}(G)$ and
$W_{def}(G)$. In particular, we show that for any $l\in \mathbb{N}$, there
exists a graph $G$ such that $def(G)>0$ and $W_{def}(G)-w_{def}(G)\geq l$. It
is known that for the complete graph $K_{2n+1}$, $def(K_{2n+1})=n$ ($n\in
\mathbb{N}$). Recently, Borowiecka-Olszewska, Drgas-Burchardt and Ha{\l}uszczak
posed the following conjecture on the deficiency of near-complete graphs: if
$n\in \mathbb{N}$, then $def(K_{2n+1}-e)=n-1$. In this paper, we confirm this
conjecture.
