**Authors:** R.R. Kamalian

### Abstract

A proper edge $t$-coloring of an undirected, simple, finite, connected graph $G$ is a coloring
of its edges with colors $1,2,...,t$ such that all colors are used, and no two adjacent edges receive the
same color. A cyclically-interval $t$-coloring of a graph $G$ is a proper edge $t$-coloring of $G$ such
that for each its vertex $x$ at least one of the following two conditions holds: a) the set of colors used
on edges incident to $x$ is an interval of integers, b) the set of colors not used on edges incident to $x$
is an interval of integers.
%A cyclically-interval $t$-coloring of a graph $G$ is a proper edge $t$-coloring of $G$ such that for each its vertex
%$x$ at least one of the following two statements is true: a) the set of colors used on edges incident to $x$ forms
%an interval of integers, b) the set of colors not used on edges incident to $x$ forms an interval of integers.
For any positive integer $t$, let $\mathfrak{M}_t$ be the set of graphs for which there exists a cyclically-interval
$t$-coloring. Examples of bipartite graphs that do not belong to the class $\bigcup\limits_{t\geq 1}\mathfrak{M}_t$
are constructed.

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