On planarity and colorability of circulant graphs

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 16 k6m}. 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 2 divides a1; : : : ; am; 2l+1|n, but 2Aaj for 16 j6m. If m6 2, we also calculate the chromatic number of G. c © 2002 Elsevier Science B.V. All rights reserved.