The Bisection Problem for Graphs of Degree 4 (Configuring Transputer Systems)

It is well known that for each k≥3 there exists such a constant c k and such an infinite sequence {Gn} ∞ n=8 of k-degree graphs (each G n has exactly n vertices) that the bisection width of G n is at least c k ·n. It this paper some upper bounds on ck's are found. Let σk(n) be the maximum of bisection widths of all k-bounded graphs of n vertices. We prove that $$\sigma _k \left( n \right) \leqslant \frac{{\left( {k - 2} \right)}}{4} \cdot n + o\left( n \right)$$ for all k=2r, r≥2. This result is improved for k=4 by constructing two algorithms A and B, where for a given 4-degree-bounded graph G n of n vertices (i) A constructs a bisection of G n involving at most n/2+4 edges for even n≤76 (i.e., σ4(n)≤n/2+4 for even n≤76) (ii) B constructs a bisection of G n involving at most n/2+2 edges for n≥256 (i.e. σ4(n)≤n/2+2 for n≥256).