Further Results on T-Coloring and Frequency Assignment Problems

A graph coloring problem (called T-coloring) is investigated in which integers are assigned to the vertices of a graph G, under the constraint that the absolute value of the difference between integers assigned to adjacent vertices does not belong to a forbidden set (called the T-set). The T-coloring problem has applications in the radio frequency channel assignment problem. T-sets of the form $\{0,s,23,\ldots,ks\}\cup S$, where $s,k\geq 1$ and $S\supseteq\{s+1,s+2,\ldots,ks-1\}$, are considered. This set has been named the k-multiple of s set. Values of interest are the minimum cardinality, $\chi_T(G)$, and the minimum span ${\rm sp}_T(G)$, of the set of assigned integers, over all possible T-colorings of G. A greedy algorithm, which T-colors perfectly orderable graphs in $\chi_T(G)$ colors with span ${\rm sp}_T(G)$ (for a k-multiple of s set), is given.