Further results on the deficiency of graphs

A \emph{proper $t$-edge-coloring} of a graph $G$ is a mapping $\alpha: E(G)\rightarrow {1,\ldots,t}$suchthatallcolorsareused,and$\alpha(e)\neq \alpha(e^{\prime})$foreverypairofadjacentedges$e,e^{\prime}\in E(G)$\mathrm{.}If$\alpha $isaproperedge-coloringofagraph$G$and$v\in V(G), then \emph{the spectrum of a vertexv}, denoted byS\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 $S(v,\alpha )$ to form an interval, and \emph{the deficiency $def(G,\alpha )$ 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: $def(G)={\min }_{\alpha }def(G,\alpha )$, 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 $W_{def}(G)-w_{def}(G)\ge l$. 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.