On planarity and colorability of circulant graphs

Authors:
Clemens Heuberger
Affiliations:
Institut für Mathematik B, Technische Universität Graz, Steyrergasse 30, Graz A-8010, Austria
Venue:
Discrete Mathematics
Year:
2003

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Abstract

For given positive integers n, a1,..., am, we consider the undirected circulant graph G = (V, E) with set of vertices V = {0,..., n - 1} and set of edges E = {[i,j]: i - j ≡ ±ak (mod n) for some 1 ≤ k ≤ m}. We prove that G is planar if m = 1 and non-planar if m ≥ 3. For m = 2 we completely characterize planarity. It is shown that G is bipartite if and only if there is an l such that 2l divides a1,..., am, 2l+1|n, but 2l+1|aj for 1 ≤ j ≤ m. If m ≤ 2, we also calculate the chromatic number of G.