Efficient channel assignment for cellular networks modeled as honeycomb grid

The channel assignment problem with separation is formulated as a vertex coloring problem of a graph G = (V,E) where each vertex represents a base station and two vertices are connected by an edge if their corresponding base stations are interfering to each other. The L(δ1, δ2, · · · , δt) coloring of G is a mapping f : V → {0, 1, · · · , λ} such that |f(u) − f(v)| ≥ δi if d(u, v) = i, where d(u, v) denotes the distance between vertices u and v in G and 1 ≤ i ≤ t. Here λ, the largest color assigned to a vertex of G, is known as the span. The same color can be reused in two vertices u and v if d(u, v) ≥ t+1, where t+1 is the reuse distance. The objective is to minimize λ over all such coloring function f . Here (δ1, δ2, · · · , δt) is called the separation vector where δ1, δ2, · · · , δt are positive integers with δ1 ≥ δ2 ≥ · · · ≥ δt. Let λ∗ be the minimum span such that there exists an L(1, 1, · · · , 1) coloring of G. We denote the separation vector (1, 1, · · · , 1) as (1). We deal with the problem of finding the maximum value of δ1 such that there exists an L(δ1, 1 t−1) coloring with span equal to λ∗. So far bounds on δ1 have been obtained for L(δ1, 1 t−1) coloring with span λ∗ for the square and triangular grids. Shashanka et al. [18] posed the problem as open for the honeycomb grid. We give lower and upper bounds of δ1 for L(δ1, 1 t−1) coloring with span λ∗ of the honeycomb grid. The bounds are asymptotically tight. We also present color assignment algorithms to achieve the lower bound.