A Variant of the Sum Coloring Problem on Trees

Let G = (V,E) be a simple and connected graph. The graph coloring problem on G is to color the vertices of G such that the colors of any two adjacent vertices are different and the number of used colors is as small as possible. By assigning a positive integer to a color, the sum coloring problem asks the minimum sum of the coloring numbers assigned for all vertices. In this paper, we introduce a new coloring problem called the distinguishable sum coloring problem. In this problem, one additional condition is required, i.e., all the vertices in the closed neighborhood of any vertex must be colored in different colors. We propose a dynamic programming algorithm for solving the distinguishable sum coloring problem on trees. The time complexity of this algorithm is O(n×Δ(T )2×Δ(T )!), where T is a tree, n = |V |, and Δ(T ) is the maximum degree of T . Also, we obtain a recurrence relation for this problem on full k-ary trees. Note that if Δ(T ) is constant, then our algorithm runs in O(n) time.