Minimum degree and size conditions for the proper connection number of graphs

An edge-coloured graph $G$ is called $properly$ $connected$ if every two vertices are connected by a proper path. The $proper$ $connection$ $number$ of a connected graph $G$, denoted by $pc(G)$, is the smallest number of colours that are needed in order to make $G$ properly connected. Susan A. van Aardt et al. gave a sufficient condition for the proper connection number to be at most $k$ in terms of the size of graphs. In this note, %optimizes the boundary of the number of edges %we study the $proper$ $connection$ $number$ is under the conditions of adding the minimum degree and optimizing the number of edges. our main result is the following, by adding a minimum degree condition: Let $G$ be a connected graph of order $n$, $k\ge 3$. If $|E(G)|\geq \binom{n-m-(k+1-m)(\delta+1)}{2} +(k+1-m)\binom{\delta+1}{2}+k+2$,then$pc(G)\leq k$,where$m$takesthevalue$t$if$\delta=1$and$\lfloor \frac{k}{\delta-1} \rfloor$if$\delta\geq2$\mathrm{.}Furthermore,if$k=2$and$\delta=2$,$|E(G)|\geq \binom{n-5}{2} +7$)$pc(G)\leq 2$,except$G\in {G_{1}, G_{n}}$($n\geq8$),where$G_{1}=K_{1}\vee 3K_{2}$and$G_{n}$ is obtained by taking a complete graph $K_{n-5}$ and $K_1\vee (2K_2$) with an arbitrary vertex of $K_{n-5}$ and a vertex with $d(v)=4$ in $K_{1}\vee (2K_{2}$)beingjoined\mathrm{.}If$k=2$,$\delta \geq 3$,weconjecture$pc(G)\leq 2$, where $m$ takes the value $1$ if $\delta =3$ and $0$ if $\delta \ge 4$ in the assumption.