Further results on the deficiency of graphs

A emph{proper $t$-edge-coloring} of a graph $G$ is a mapping $alpha: E(G)rightarrow {1,ldots,t}$ such that all colors are used, and $alpha(e) eq alpha(e^{prime})$ for every pair of adjacent edges $e,e^{prime}in E(G)$. If $alpha $ is a proper edge-coloring of a graph $G$ and $vin V(G)$, then emph{the spectrum of a vertex $v$}, denoted by $Sleft(v,alpha right)$, is the set of all colors appearing on edges incident to $v$. emph{The deficiency of $alpha$ at vertex $vin V(G)$}, denoted by $def(v,alpha)$, is the minimum number of integers which must be added to $Sleft(v,alpha right)$ to form an interval, and emph{the deficiency $defleft(G,alpharight)$ of a proper edge-coloring $alpha$ of $G$} is defined as the sum $sum_{vin V(G)}def(v,alpha)$. emph{The deficiency of a graph $G$}, denoted by $def(G)$, is defined as follows: $def(G)=min_{alpha}defleft(G,alpharight)$, 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 $lin mathbb{N}$, 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 $K_{2n+1}$, $def(K_{2n+1})=n$ ($nin mathbb{N}$). Recently, Borowiecka-Olszewska, Drgas-Burchardt and Ha{l}uszczak posed the following conjecture on the deficiency of near-complete graphs: if $nin mathbb{N}$, then $def(K_{2n+1}-e)=n-1$. In this paper, we confirm this conjecture.