Vertex-magic total labelings of even complete graphs

It is shown that if p ≥ 6 is any even integer such that p ≡ 2 mod ( 4 ) then the complete graph K p has a vertex-magic total labeling (VMTL) with magic constant h for each integer h satisfying p 3 + 6 p ≤ 4 h ≤ p 3 + 2 p 2 − 2 p . If in addition, p ≡ 2 mod ( 8 ) , then K p has a VMTL with magic constant h for each integer h satisfying p 3 + 4 p ≤ 4 h ≤ p 3 + 2 p 2 . If p = 2 ⋅ 3 t and t ≥ 2 , then it is shown that the complete graph K p has a VMTL with magic constant h if and only if h is an integer satisfying p 3 + 3 p ≤ 4 h ≤ p 3 + 2 p 2 + p . These results provide significant new evidence supporting a conjecture of MacDougall, Miller, Slamin and Wallis regarding the spectrum of complete graphs. It is also shown that for each odd integer n ≥ 5 , the disjoint union of two copies of K n , denoted 2 K n , has a VMTL with magic constant h for each integer h such that n 3 + 5 n ≤ 2 h ≤ n 3 + 2 n 2 − 3 n . If in addition, n ≡ 1 mod 4 , then 2 K n has a VMTL with magic constant h for each integer h such that n 3 + 3 n ≤ 2 h ≤ n 3 + 2 n 2 − n .