One Curious Identity Counting Graceful Labelings

TL;DR

This paper establishes a precise formula for counting essentially distinct alpha-graceful labelings of certain complete bipartite graphs, revealing a surprising connection to alternating sums of binomial coefficient powers.

Contribution

It provides a novel exact enumeration formula for alpha-graceful labelings of specific bipartite graphs, linking graph labelings to binomial coefficient sums.

Findings

Number of labelings equals an alternating sum of fourth powers of binomial coefficients.

The formula applies to graphs with prime factorization structures as specified.

Establishes a new combinatorial identity involving graph labelings and binomial sums.

Abstract

Let $a$ and $b$ be positive integers with prime factorisations $a = p_1^np_2^n$ and $b = q_1^nq_2^n$. We prove that the number of essentially distinct $\alpha$-graceful labelings of the complete bipartite graph $K_{a, b}$ equals the alternating sum of fourth powers of binomial coefficients $(-1)^n[\binom{2n}{0}^4 - \binom{2n}{1}^4 + \binom{2n}{2}^4 - \binom{2n}{3}^4 + \cdots + \binom{2n}{2n}^4]$.