Magic rectangles revisited

Magic Graphs

“…It was shown in [43,44] that such an array exists whenever m and n have the same parity, except for the impossible cases where exactly one of m and n is 1, and for m = n = 2. Simpler constructions appear in [42].…”

“…For our purposes we need the existence of a 3×m magic rectangle for every odd m, so for completeness we present constructions (from [42]) for this case. The construction uses smaller arrays as building blocks.…”

“…In the case where m = 1 we have (23,2,26,3,31,4,22,5,25,8) as the vertex labels on the cycle with k = 46. When m = 2 we have (38,6,39,9,40,2,41,5,42,13,444,7,45,8,46,4,55,3) as the vertex labels on the cycle with k = 78. For m ≥ 3 we have k = 32m + 14 and define…”

“…Universal labelings of C5 and C9 Theorem 2 42. The following vertex labeling λ induces a super edge-magic labeling of C t 4m+1 for every t except t = 5, 9, 4m − 4, 4m − 8, other than m ≥ 3:…”

Numbers of edges in supermagic graphs

“…A part of definition of g was inspired by [4]. It is easy to see that the mapping g is a bijection and for its index-mapping we get…”