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**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 \(T\).

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

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\textit{J. H. Rabinowitz} and \textit{V. K. Proulx}, SIAM J. Algebraic Discrete Methods 6, 507--518 (1985; Zbl 0579.05058)

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### References:

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