On the chromatic index of graphs with 2 m + 1 vertices and 2 m2 edges

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Consider the chromatic index problem for K 2 m + 1 , the complete graphs with 2 m + 1 vertices, m = 1,2,…. The chromatic index of these graphs is known to be 2 m + 1, the degree of the graphs plus one. Plantholt proved that K 2 m + 1 − m, the graph obtained after removing arbitrary m edges from K 2 m + 1 , has the chromatic index 2 m. In this paper, we present a much simpler constructive proof for this problem as well as an O( m 2) optimal time algorithm for actually assigning colors to the edges.