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For an undirected, simple, finite, connected graph $G$, we denote by $V(G)$and $E(G)$ the sets of its vertices and edges, respectively. A function$\varphi:E(G)\rightarrow \{1,...,t\}$ is called a proper edge $t$-coloring of agraph $G$, if adjacent edges are colored differently and each of $t$ colors isused. The least value of $t$ for which there exists a proper edge $t$-coloringof a graph $G$ is denoted by $\chi'(G)$. For any graph $G$, and for any integer$t$ satisfying the inequality $\chi'(G)\leq t\leq |E(G)|$, we denote by$\alpha(G,t)$ the set of all proper edge $t$-colorings of $G$. Let us alsodefine a set $\alpha(G)$ of all proper edge colorings of a graph $G$: $$\alpha(G)\equiv\bigcup_{t=\chi'(G)}^{|E(G)|}\alpha(G,t). $$  An arbitrary nonempty finite subset of consecutive integers is called aninterval. If $\varphi\in\alpha(G)$ and $x\in V(G)$, then the set of colors ofedges of $G$ which are incident with $x$ is denoted by $S_G(x,\varphi)$ and iscalled a spectrum of the vertex $x$ of the graph $G$ at the proper edgecoloring $\varphi$. If $G$ is a graph and $\varphi\in\alpha(G)$, then define$f_G(\varphi)\equiv|\{x\in V(G)/S_G(x,\varphi) \textrm{is an interval}\}|$.  For a graph $G$ and any integer $t$, satisfying the inequality $\chi'(G)\leqt\leq |E(G)|$, we define: $$\mu_1(G,t)\equiv\min_{\varphi\in\alpha(G,t)}f_G(\varphi),\qquad\mu_2(G,t)\equiv\max_{\varphi\in\alpha(G,t)}f_G(\varphi). $$  For any graph $G$, we set: $$ \mu_{11}(G)\equiv\min_{\chi'(G)\leqt\leq|E(G)|}\mu_1(G,t),\qquad \mu_{12}(G)\equiv\max_{\chi'(G)\leqt\leq|E(G)|}\mu_1(G,t), $$ $$ \mu_{21}(G)\equiv\min_{\chi'(G)\leqt\leq|E(G)|}\mu_2(G,t),\qquad \mu_{22}(G)\equiv\max_{\chi'(G)\leqt\leq|E(G)|}\mu_2(G,t). $$  For any positive integer $n$, the exact values of the parameters $\mu_{11}$,$\mu_{12}$, $\mu_{21}$ and $\mu_{22}$ are found for the graph of the$n$-dimensional cube.

On the extremal values of the number of vertices with an interval  spectrum on the set of proper edge colorings of the graph of the  $n$-dimensional cube