Interval Edge Coloring of Trees with Strict Restrictions on the Spectrums

An edge-coloring of a graph G with consecutive integers C1 ,..., Ct is called an interval t-coloring if all the colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. A graph G is interval colorable if it has an interval t-coloring for some positive integer t. For an edge coloring a and a vertex v the set of all the colors of the incident edges of v is called the spectrum of that vertex in a and is denoted by Sa(v). We consider the case where the spectrum for each vertex v is provided S(v), and the problem is to find an edge-coloring a such that for every vertex v, Sa(v)=S(v). We provide an O(N) algorithm that finds such an edge-coloring for trees that satisfies all the restrictions. If it is impossible to have an edge-coloring that satisfies the restrictions of the spectrums the algorithm will tell that too