Interval colorings of edges of a multigraph

Let G = (V (G), E(G)) be a multigraph. The degree of a vertex x in G is denoted by d(x), the greatest degree of a vertex – by ∆(G), the chromatic index of G – by χ ′ (G). Let R ⊆ V (G). An interval (respectively, continuous) on R t-coloring of a multigraph G is a proper coloring of edges of G with the colors 1, 2,. .. , t, in which each color is used at least for one edge, and the edges incident with each vertex x ∈ R are colored by d(x) consecutive colors (respectively, by the colors 1, 2,. .. , d(x)). In this paper the problems of existence and construction of interval or continuous on R colorings of G are investigated. Problems of such kind appear in construction of timetablings without "windows". Some properties of interval or continuous on V (G) colorings were obtained in [1, 2]. Necessary and sufficient conditions of the existence of a continuous on V (G) ∆(G)-coloring in the case when G is a tree are obtained in [3]. All non-defined concepts can be found in [4, 5]. Let N t be the set of multigraphs G, for which there exists an interval on V (G) t-coloring, and N = t≥1 N t. For every G ∈ N, let us denote by w(G) and W (G), respectively, the least and the greatest t, for which there exists an interval on V (G) t-coloring of G. Evidently, ∆(G) ≤ χ ′ (G) ≤ w(G) ≤ W (G).