One Curious Identity Counting Graceful Labelings

Nikolai Beluhov

doi:10.54550/ECA2022V2S1R6

paper

One Curious Identity Counting Graceful Labelings

arXiv:2102.00412 · doi:10.54550/ECA2022V2S1R6

Abstract

Let $a$ and $b$ be positive integers with prime factorisations $a = p_{1}^{n} p_{2}^{n}$ and $b = q_{1}^{n} q_{2}^{n}$ . We prove that the number of essentially distinct $α$ -graceful labelings of the complete bipartite graph $K_{a, b}$ equals the alternating sum of fourth powers of binomial coefficients $(- 1)^{n} [(0 2 n)^{4} - (1 2 n)^{4} + (2 2 n)^{4} - (3 2 n)^{4} + \dots + (2 n 2 n)^{4}]$ .

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