On cyclically-interval edge colorings of trees

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 $\phi :E(G)\to \{1,2,\dots ,t\}$ is called a proper edge $t$-coloring of a graph $G$ if adjacent edges are colored differently and each of $t$ colors is used. An arbitrary nonempty subset of consecutive integers is called an interval. If $\phi$ is a proper edge $t$-coloring of a graph $G$ and $x\in V(G)$,then$S_G(x,\varphi)$denotesthesetofcolorsofedgesof$G$ which are incident with $x$. A proper edge $t$-coloring $\phi$ of a graph $G$ is called a cyclically-interval $t$-coloring if for any $x\in V(G)$ at least one of the following two conditions holds: a) $S_G(x,\phi )$ is an interval, b) $\{1,2,\dots ,t\}\setminus S_G(x,\phi )$ is an interval. For any $t\in $\mathbb{N}$$,let$\mathfrak{M}_t$ be the set of graphs for which there exists a cyclically-interval $t$-coloring, and let $$\mathfrak{M}\equiv \bigcup _{t\ge 1}{\mathfrak{M}}_t\mathrm{.}$$ For an arbitrary tree $G$, it is proved that $G\in \mathfrak{M}$ and all possible values of $t$ are found for which $G\in {\mathfrak{M}}_t\mathrm{.}$