Abstract: The graph composed of k subgraphs, each the totally-disconnected graph with n nodes, arranged in a cycle, can be edge-colored with 2n colors unless n and k are both odd. The edge coloring number of a graph is the smallest number of colors for its edges so that no two adjacent edges have like color. Vizing [1] showed that the edge coloring number is either the maximum degree of a node, or one more than this number. In this note we consider the edge coloring number of a graph composed of k copies of the totally disconnected graph with n nodes. The totally disconnected graphs are arranged in a k-cycle; i.e., two nodes are joined by an edge if and only if they are in components adjacent in the k-cycle. We show that the edge coloring number of the graph is 2n unless n and k are both odd. For n and k both odd, assume that 2n colors suffice to edge-color the graph. Then each node, being of degree 2n, has an adjacent edge of each color. Each edge is adjacent to two nodes, thus the edges of a given color must be adjacent to an even number of nodes. The graph has kn nodes, an odd number, giving a contradiction. For k even a construction of the pattern of colors is given. Designate the nodes in the first totally disconnected component by AO *, A,,-,; those in the next component in the cycle by similarly subscripted B's, etc. Call the colors XO, * * *, Xn-l and YO, * * *, Yn_-. Color edge AiB3 with ,+j, the latter subscript reduced mod n. Color edge BiCj with Y+j. Continue the alternation of X and Y around the cycle of even length k. When n is even the constructive procedure developed below yields an edge coloring. Again, proceeding around the cycle, designate nodes by subscripted A's, B's, * , etc. Let the colors be W, X, Y, and Z, each letter assigned n/2 different subscripts. Describe the colorings of the AiB, Received by the editors April 14, 1972. AMS (MOS) subject classfltcations (1970). Primary 05C99.