Interference-Minimizing Colorings of Regular Graphs

Abstract

Communications problems that involve frequency interference, such as the channel assignment problem in the design of cellular telephone networks, can be cast as graph coloring problems in which the frequencies (colors) assigned to an edge's vertices interfere if they are too similar. The paper considers situations modeled by vertex-coloring d-regular graphs with n vertices using a color set 1, 2,..., n, where colors i and j are said to interfere if their circular distance $\min \{ | i-j | , n- | i-j | \}$ does not exceed a given threshold value $\alpha$. Given a d-regular graph G and threshold $\alpha$, an interference-minimizing coloring is a coloring of vertices that minimizes the number of edges that interfere. Let $I_\alpha (G)$ denote the minimum number of interfering edges in such a coloring of $G$. For most triples $(n, \alpha ,d),$ we determine the minimum value of $I_\alpha (G)$ over all d-regular graphs and find graphs that attain it. In determining when this minimum value is 0, we prove that for $r \geq 3$ there exists a d-regular graph G on n vertices that is r-colorable whenever $d \leq (1- \frac{1}{r}) n-1$ and nd is even. We also study the maximum value of $I_\alpha (G)$ over all d-regular graphs and find graphs that attain this maximum in many cases.