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36 2 Edge-Magic Total Labelings In Fig. 4 we present two examples, for v = 8 and 10. 20 yield k = 22 and 27. 5 1 12 1 7 12 4 2 6 15 14 6 3 k = 22 2 5 13 4 3 k = 27 Fig. 4. 21 Every cycle of length divisible by 4 has an edge-magic total labeling with k = 3v. Proof. 20. So assume v ≥ 8, write v = 4m, m > 1. The required labeling is ⎧ i ⎪ ⎪ ⎪ ⎪ 4m + i + 1 ⎪ ⎪ ⎨ i+1 λ(ui ) = 4m + i ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎩ 2v − 2 for i = 1, 3, . . , 2m − 1 for i = 2, 4, . . , 2m − 2 for i = 2m, 2m + 2, . . , 4m − 2 for i = 2m + 1, 2m + 3, .

Radar distance ranging is accomplished by transmitting a pulse or train of pulses and waiting for its return after reflection. Only a small fraction of the transmitted energy ever returns to the detector. Because accuracy in measuring target distance is determined by accuracy in measuring the time until the reflected signal is received, it is desirable to have a very narrow transmitted radar pulse whose moment of return can be measured precisely. High-energy pulses are broader than low-energy pulses.

Write d = 2r s + 1 where s is an odd positive integer and r is a positive integer. If G is edge-magic, then 2r+2 divides v. Proof. As usual, 2e = dv. Since e is even and d is odd, 4 divides v; write v = 4V , so that e = 2dV . 3) becomes (4V + 2dV )(4V + 2dV + 1)/2 = 2dV k − (d − 1) λ(x), x and V [(d + 2)(4V + 2dV + 1) − 2dk] = 2r s λ(x). x The coefficient of V on the left-hand side is odd, so 2r divides V , giving the result. 22 2 Edge-Magic Total Labelings Remark. This theorem could also be expressed as follows.