Vertex-magic total labeling of odd complete graphs

Let n be an odd positive integer. It is shown that the complete graph Kn has a vertex magic total labeling with magic constant h if and only if (n/4)(n2+3)⩽h⩽(n/4)(n+1)2. This partially solves the conjecture regarding possible magic constants of complete graphs. In addition, new techniques are introduced to allow for the simultaneous construction of labelings with different magic constants, and to then alter some of these labelings to obtain new ones with slightly lesser magic constants.