On cyclically-interval edge colorings of trees

Abstract

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 $\varphi:E(G)\rightarrow\{1,2,\ldots,t\}$ 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 $\varphi$ is a proper edge $t$-coloring of a graph $G$ and $x\in V(G)$, then $S_G(x,\varphi)$ denotes the set of colors of edges of $G$ which are incident with $x$. A proper edge $t$-coloring $\varphi$ 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) $S_G(x,\varphi)$ is an interval, b) $\{1,2,\ldots,t\}\setminus S_G(x,\varphi)$ 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 $$\mathfrak{M}\equiv\bigcup_{t\geq1}\mathfrak{M}_t.$$ For an arbitrary tree $G$, it is proved that $G\in\mathfrak{M}$ and all possible values of $t$ are found for which $G\in\mathfrak{M}_t.$