Vertex-magic total labelings of even complete graphs

It is shown that if p≥6p≥6 is any even integer such that p≡2mod(4) then the complete graph KpKp has a vertex-magic total labeling (VMTL) with magic constant hh for each integer hh satisfying p3+6p≤4h≤p3+2p2−2pp3+6p≤4h≤p3+2p2−2p. If in addition, p≡2mod(8), then KpKp has a VMTL with magic constant hh for each integer hh satisfying p3+4p≤4h≤p3+2p2p3+4p≤4h≤p3+2p2. If p=2⋅3tp=2⋅3t and t≥2t≥2, then it is shown that the complete graph KpKp has a VMTL with magic constant hh if and only if hh is an integer satisfying p3+3p≤4h≤p3+2p2+pp3+3p≤4h≤p3+2p2+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≥5n≥5, the disjoint union of two copies of KnKn, denoted 2Kn2Kn, has a VMTL with magic constant hh for each integer hh such that n3+5n≤2h≤n3+2n2−3nn3+5n≤2h≤n3+2n2−3n. If in addition, n≡1mod4, then 2Kn2Kn has a VMTL with magic constant hh for each integer hh such that n3+3n≤2h≤n3+2n2−nn3+3n≤2h≤n3+2n2−n.