Max-cut in circulant graphs

Poljak, S. and D. Turzik, Max-cut in circulant graphs, Discrete Mathematics 108 (1992) 379-392. We study the max-cut problem in circulant graphs C,,,, where C,,, 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 r-regular cut. The number of edges of a t-regular cut can easily be computed. This gives an O(r log* n) 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. 1. The construction of maximum cut The circulant C,,, is the graph on the vertex set V = { 1, . . . , n}, and with the edges (i, i + 1) and (i, i + r) for i = 1, . . . , n, where i + 1 and i + r are taken modulo II. The edges (i, i + 1) and (i, i + r) are called outer and inner, respectively. We will consider only the circulants where every vertex has degree four, i.e., we assume that 12 > 2r > 3. A cut in a graph G = (V, E) is an edge set 6s : = { ij E E ( i E S, j $ S} for some vertex set S c V, 0 # S # V. The cardinality of 6s is called the size of the cut, and a cut of maximum size is said to be maximum. The max-cut problem is well-studied both in combinatorial optimization and graph theory, see e.g. [2-4, 7-131. The max-cut problem is known to be NP-complete [6]. In this paper we present an O(r log2 n) algorithm to compute the size of a maximum cut for a circulant C,,,. We explain the main idea of our result. We prove that any maximum cut &S must intersect any cycle {i, i + 1, . . . , i + I}, i=l . . 9 n, in almost the same number of edges. More precisely, the cardina&ies of intersections (which are always even) may differ at most by two. 0012-365X/92/$05.00