Circulant graphs: recognizing and isomorphism testing in polynomial time

Abstract

Abstract. An algorithm is constructed for recognizing the circulant graphs and finding a canonical labeling for them in polynomial time. This algorithm also yields a cycle base of an arbitrary solvable permutation group. The consistency of the algorithm is based on a new result on the structure of Schur rings over a finite cyclic group. §1. Introduction A finite graph1 is said to be circulant if its automorphism group contains a full cycle, i.e., a permutation the cycle decomposition of which consists of a unique cycle of full length. This means that the graph admits a regular cyclic automorphism group, and, consequently, is isomorphic to a Cayley graph over a cyclic group. In particular, any circulant graph can be specified in a compact form by a full cycle automorphism and a neighborhood of some vertex. One of the main computational problems concerning circulant graphs is that of finding an efficient algorithm to recognize them. (This problem is a special case of the following NP-complete problem: test whether or not a given graph has an automorphism without fixed points [15].) The first attempt to solve this problem was undertaken in [24], where a polynomial-time algorithm for recognizing circulant tournaments was described. In the subsequent papers [21,22,5] several results on recognizing some special classes of circulant graphs were presented, but the general problem remained open up to now. In the present paper we solve this problem completely. Another problem about circulant graphs is to find an efficient isomorphism test for them. In fact, this problem is polynomial-time reducible to the recognition problem, because two circulant graphs with the same number of vertices are isomorphic if and only if their disjoint union is a circulant graph. In this paper we present a solution to a more difficult problem of finding a canonical labeling for circulant graphs.3 It should be mentioned that the isomorphism problem for Cayley graphs over a cyclic group (which is a special case of the isomorphism problem for circulant graphs) has been extensively studied through the last forty years (see [20]

Powers of cycles, powers of paths, and distance graphs

“…If D satisfies (12) for some p, k ∈ N and 1 + (k − 1)p ≤ n−1 2 , then D and an ordering v 0 , v 1 , . .…”

“…Circulant graphs are the Cayley graphs of cyclic groups and due to their symmetry and connectivity properties, they have been proposed for various practical applications [1]. Isomorphism testing and recognition of circulant graphs had been long-standing open problems [23][24][25] and were completely solved only recently [12,22].…”

“…Distance graphs lack the symmetry of circulant graphs and the algebraic methods used in [12,22] do not apply to them. At first sight they seem to generalize powers of paths in a similar way as circulant graphs generalize powers of cycles.…”

Testing Isomorphism of Central Cayley Graphs Over Almost Simple Groups in Polynomial Time

“…Theorem 4.4. Let G be an almost simple group, K a 2-closed group satisfying condition (6), and U, L are the partitions associated with the principal K-section. Then the coherent configuration Inv(K L ) is the direct sum of the coherent configu-…”

Schur rings over a Galois ring of odd characteristic

“…Man proved that any Schur ring (S-ring) over a finite cyclic group can be constructed from special S-rings by means of two operations: tensor product and wedge product (as for a background of S-rings see Section 7). This theorem supplemented with the normality theory from [5] enabled to get a series of strong results in algebraic combinatorics [5,6,10,14].…”