A proper edge-coloring of a graph $G$ with colors $1,ldots,t$ is called an emph{interval cyclic $t$-coloring} if all colors are used, and the edges incident to each vertex $vin V(G)$ are colored by $d_{G}(v)$ consecutive colors modulo $t$, where $d_{G}(v)$ is the degree of a vertex $v$ in $G$. A graph $G$ is emph{interval cyclically colorable} if it has an interval cyclic $t$-coloring for some positive integer $t$. The set of all interval cyclically colorable graphs is denoted by $mathfrak{N}_{c}$. For a graph $Gin mathfrak{N}_{c}$, the least and the greatest values of $t$ for which it has an interval cyclic $t$-coloring are denoted by $w_{c}(G)$ and $W_{c}(G)$, respectively. In this paper we investigate some properties of interval cyclic colorings. In particular, we prove that if $G$ is a triangle-free graph with at least two vertices and $Gin mathfrak{N}_{c}$, then $W_{c}(G)leq vert V(G)vert +Delta(G)-2$. We also obtain bounds on $w_{c}(G)$ and $W_{c}(G)$ for various classes of graphs. Finally, we give some methods for constructing of interval cyclically non-colorable graphs.
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