An Exact Algorithm for the Generalized List TT-Coloring Problem

The generalized list $T$-coloring is a common generalization of many graph coloring models, including classical coloring, $L(p,q)$-labeling, channel assignment and $T$-coloring. Every vertex from the input graph has a list of permitted labels. Moreover, every edge has a set of forbidden differences. We ask for such a labeling of vertices of the input graph with natural numbers, in which every vertex gets a label from its list of permitted labels and the difference of labels of the endpoints of each edge does not belong to the set of forbidden differences of this edge. In this paper we present an exact algorithm solving this problem, running in time $\mathcal{O}^*((\tau+2)^n)$, where $\tau$ is the maximum forbidden difference over all edges of the input graph and $n$ is the number of its vertices. Moreover, we show how to improve this bound if the input graph has some special structure, e.g. a bounded maximum degree, no big induced stars or a perfect matching

Distance Two Labeling of Generalized Cacti

In a communication network, the main task is to assign a channel (non negative integer) to each TV or radio transmitters located at different places such that communication do not interfere. This problem is known as channel assignment problem which was introduced by Hale [4]. Usually, the interference between two transmitters is closely related with the geographic location of the transmitters. If we consider two level interference namely major and minor then two transmitters are very close if the interference is major while close if the interference is minor. Robert [7] proposed a variation of the channel assignment problem in which close transmitters must receive different channels and very close transmitters must receive channels that are at two apart