Some remarks on relations between the $\mu$-parameters of regular graphs

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 $\phi :E(G)\to \{1,\mathrm{.}\mathrm{.}\mathrm{.},t\}$ is called a proper edge $t$-coloring of a graph $G$, if adjacent edges are colored differently and each of $t$ colors is used. The least value of $t$ for which there exists a proper edge $t$-coloring of a graph $G$ is denoted by ${\chi }^{\mathrm{\prime }}(G)$. For any graph $G$, and for any integer $t$ satisfying the inequality ${\chi }^{\mathrm{\prime }}(G)\le t\le \mathrm{\mid }E(G)\mathrm{\mid }$, we denote by $\alpha (G,t)$ the set of all proper edge $t$-colorings of $G$. Let us also define a set $\alpha (G)$ of all proper edge colorings of a graph $G$: $$\alpha (G)\equiv \bigcup _{t={\chi }^{\mathrm{\prime }}(G)}^{\mathrm{\mid }E(G)\mathrm{\mid }}\alpha (G,t)\mathrm{.}$$ An arbitrary nonempty finite subset of consecutive integers is called an interval. If $\phi \in \alpha (G)$ and $x\in V(G)$, then the set of colors of edges of $G$ which are incident with $x$ is denoted by $S_G(x,\phi )$ and is called a spectrum of the vertex $x$ of the graph $G$ at the proper edge coloring $\phi$. If $G$ is a graph and $\phi \in \alpha (G)$, then define $f_G(\phi )\equiv \mathrm{\mid }\{x\in V(G)\mathrm{/}{S}_G(x,\phi )\text{is an interval}\}\mathrm{\mid }$. For a graph $G$ and any integer $t$, satisfying the inequality $\chi’(G)\leq t\leq |E(G)|$,wedefine:$$ \mu_1(G,t)\equiv\min_{\varphi\in\alpha(G,t)}f_G(\varphi),\qquad \mu_2(G,t)\equiv\max_{\varphi\in\alpha(G,t)}f_G(\varphi). For any graph $G$, we set: \mu_{11}(G)\equiv\min_{\chi’(G)\leq t\leq|E(G)|}\mu_1(G,t),\qquad \mu_{12}(G)\equiv\max_{\chi’(G)\leq t\leq|E(G)|}\mu_1(G,t), $$ $${\mu }_{21}(G)\equiv \underset{{\chi }^{\mathrm{\prime }}(G)\le t\le \mathrm{\mid }E(G)\mathrm{\mid }}{\min }{\mu }_2(G,t),{\mu }_{22}(G)\equiv \underset{{\chi }^{\mathrm{\prime }}(G)\le t\le \mathrm{\mid }E(G)\mathrm{\mid }}{\max }{\mu }_2(G,t)\mathrm{.}$$ For regular graphs, some relations between the $\mu$-parameters are obtained.