We study the structural constraint of random scale-free networks that determines possible combinations of the degree exponent gamma and the upper cutoff k(c) in the thermodynamic limit. We employ the framework of graphicality transitions proposed by Del Genio and co-workers [Phys. Rev. Lett. 107, 178701 (2011)], while making it more rigorous and applicable to general values of k(c). Using the graphicality criterion, we show that the upper cutoff must be lower than k(c) similar to N-1/gamma for gamma < 2, whereas any upper cutoff is allowed for gamma > 2. This result is also numerically verified by both the random and deterministic sampling of degree sequences.