## An asymptotic approach to the channel assignment problem.(English)Zbl 0579.05058

Let $$T$$ be a nonempty finite set of positive integers, and let $$G$$ be a finite simple graph. A $$T$$-coloring $$f$$ of $$G$$ is an assignment of an integer $$f(v)$$ to each vertex $$v$$ of $$G$$ such that $$| f(v)-f(w)| \not\in T\cup \{0\}$$ whenever $$v$$ and $$w$$ are adjacent in $$G$$. The span of $$f$$ is the difference between the largest and smallest integers assigned, and the $$T$$-span of $$G$$ is the minimum span over all possible assignments $$f$$. (The problem arises from the formulation by W. K. Hale of the channel assignment problem for potentially interfering communication nets.)
Let $$s(T,n)$$ denote the $$T$$-span of the complete graph $$K_ n$$ on $$n$$ vertices. It is shown that there is a “rate function” $$r$$ from the set of all finite nonempty sets $$T$$ of positive integers to the set of positive rationals not greater than $$1/2$$, such that as $$n\to \infty$$ have $$\lim (n/s(T,n))=r(T)$$. Moreover, a finite algorithm is given for computing $$r(T)$$; the algorithm is exponential in the largest element of $$T$$, but allows $$K_ n$$ to be $$T$$-colored in an “asymptotically optimal” way in constant time (as a function of $$n$$). The output of the algorithm is an infinite sequence of integers (generated from a certain previously computed periodic sequence of zeros and ones), with the property that its first n elements may be used to $$T$$-color $$K_ n$$ so that the resulting span differs from $$n/r(T)$$ by no more than some fixed integer; this coloring is thus “asymptotically optimal” in the sense that the error $$e(n)$$ between its span and $$s(T,n)$$ has the property that $$e(n)/n\to 0$$ as $$n\to \infty.$$
Some properties of $$r(T)$$ are developed, in terms of the structure of the set $$T$$. These include general bounds for $$r(T)$$, and exact values for certain special sets $$T$$.
Reviewer: T. D. Parsons

### MSC:

 05C99 Graph theory 05C15 Coloring of graphs and hypergraphs
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### References:

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