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Max-cut in circulant graphs. (English) Zbl 0769.05059
Summary: We study the max-cut problem in circulant graphs $$C_{n,r}$$, where $$C_{n,r}$$ is a graph whose edge set consists of a cycle of length $$n$$ and all the vertex pairs of distance $$r$$ on the cycle. An efficient solution of the problem is obtained so that we show that there is always a maximum cut of a particular shape, called a $$t$$-regular cut. The number of edges of a $$t$$-regular cut can easily be computed. This gives an $$O(r\log^ 2n)$$ time algorithm for the max-cut.
We present also some new classes of facets of the bipartite subgraph polytope and the cut polytope, which are spanned by $$t$$-regular cuts.

##### MSC:
 05C38 Paths and cycles 05C85 Graph algorithms (graph-theoretic aspects) 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) 05C99 Graph theory 68R10 Graph theory (including graph drawing) in computer science
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