Interval colorings of edges of a multigraph

Let $G=(V_1(G),V_2(G),E(G))$ be a bipartite multigraph, and $R\subseteq V_1(G)\cup V_2(G)$\mathrm{.}Apropercoloringofedgesof$G$withthecolors$1,\ldots,t$iscalledinterval(respectively,continuous)on$R$, if each color is used for at least one edge and the edges incident with each vertex $x\in R$ are colored by $d(x)$ consecutive colors (respectively, by the colors $1,\dots ,d(x))$, where $d(x)$ is a degree of the vertex $x$. We denote by $w_1(G)$ and $W_1(G)$, respectively, the least and the greatest values of $t$, for which there exists an interval on $V_1(G)$ coloring of the multigraph $G$ with the colors $1,\dots ,t$. In the paper the following basic results are obtained. $\mathbf{\text{Theorem}}\mathbf{\text{2.}}$ For an arbitrary $k$, $w_1(G)\le k\le W_1(G)$, there is an interval on $V_1(G)$ coloring of the multigraph $G$ with the colors $1,\dots ,k$. $\mathbf{\text{Theorem}}\mathbf{\text{3.}}$ The problem of recognition of the existence of a continuous on $V_1(G)$ coloring of the multigraph $G$ is $NP$-complete. $\mathbf{\text{Theorem}}\mathbf{\text{4.}}$ If for any edge $(x,y)\in E(G)$, where $x\in V_1(G)$, the inequality $d(x)\ge d(y)$ holds then there is a continuous on $V_1(G)$ coloring of the multigraph $G$. $\mathbf{\text{Theorem}}\mathbf{\text{1.}}$ If $G$ has no multiple edges and triangles, and there is an interval on $V(G)$ coloring of the graph $G$ with the colors $1,\dots ,k$, then $k\le \mathrm{\mid }V(G)\mathrm{\mid }-1$.