GRACEFUL NUMBERS

We construct a labeled graph D(n) that reﬂects the structure of divisors of a given natural number n . We deﬁne the concept of graceful numbers in terms of this associated graph and ﬁnd the general form of such a number. As a consequence, we determine which graceful numbers are perfect.


Introduction.
In [2], Gallian presented a detailed survey of various types of graph labeling, the two best known being graceful and harmonious. Recall that a graph G with q edges is called graceful if one can label its vertices with distinct numbers from the set {0, 1,...,q} and mark the edges with differences of the labels of the end vertices in such a way that the resulting edge labels are distinct. A number of interesting results on graceful and graceful-like labelings are obtained in [1,3,4] and some other works. In this note, we give a description of natural numbers whose associated graph of divisors satisfies certain graceful-like conditions. For any natural number n, we construct a labeled graph D(n) that reflects the structure of divisors of n. We define the concept of graceful number in terms of this associated graph and find the general form of such a number. As a consequence, we determine which graceful numbers are perfect.

Main results.
Given a natural number n one can generate a graph D(n) that reflects the structure of divisors of n as follows. The vertices of the graph represent all the divisors of the number n, each vertex is labeled by a certain divisor. (In what follows, we refer to the vertex of the graph D(n) with label k as the "vertex k.") If r and s are two divisors of n and r > s, then there is an edge between the vertices s and r if and only if s divides r and the ratio r /s is a prime number. As in the theory of graceful graphs, we label such an edge by the difference r − s of the labels of its vertices. In what follows, the sum of the labels of all edges of the graph D(n) is denoted by SD(n) while SD(n) denotes the sum of labels of all edges of D(n) except the edges terminating at n. (Clearly, if n = p  p ··· The following example shows that there are numbers n such that SD(n) > n, as well as numbers that satisfy the condition SD(n) = n.
In order to obtain the description of graceful numbers, we first find the value of SD(n) when n is a product of powers of two different prime numbers.  and (the first sum corresponds to the differences of the consecutive divisors of n when the exponent of q decreases, and the second sum takes care about the differences of consecutive divisors of n when the exponent of p decreases). Thus, It follows from formulas (2.1) and (2.2) that a number n = p r q s (p and q are prime, r ≥ 1, and s ≥ 1) is graceful if and only if p = 2 and s = 1, that is, n = 4q for some odd prime number q.
Indeed, equality SD(n) = n can hold only for even numbers n (if n is odd, then (2.1) shows that SD(n) is even, whence SD(n) ≠ n). If n = 2 r q s , where r ≥ 2, s ≥ 2, then , so that SD(2 r q) = 2 r q if and only if r = 2. Thus, for any two different prime numbers p and q, p < q, and for any two nonnegative integers r and s, the number p r q s is graceful if and only if p = 2, r = 2, and s = 1. Now, we generalize formula (2.1) to the case of arbitrary number n. More precisely, we show that if n = p We proceed by induction on n. We have seen that the formula is true if n is a power of a prime number or a product of two powers of primes. In order to perform the step of induction, notice that Applying the inductive hypothesis and taking into account that we obtain that Formula (2.8) shows, in particular, that if a number n is odd, then SD(n) is even (it is easily seen that both sums in the right side of the formula are even if n is odd). Therefore, every graceful number must be even, that is, for some odd primes q 1 ,...,q m (m ≥ 1,s i ≥ 1 for i = 1,...,m). As we have seen, if m = 1, then the number n is graceful if and only if s 1 = 1 and r = 2, that is, n = 4q 1 .
We show that if m ≥ 2, then SD(n) > n, so the only graceful numbers are the numbers of the form 4q where q is an odd prime. First of all, notice that SD(2 r q s 1 1 ) ≥ 2 r q s 1 1 for r ≥ 1, s ≥ 2 (see Example 2.4) and SD(2q 1 q 2 ) ≥ 2q 1 q 2 for any two different primes q 1 and q 2 (applying formula (2.1) we obtain that SD(2q 1 q 2 ) = (q 1 + 1)(q 2 + 1) + 3(q 1 − 1)(q 2 + 1) + 3(q 2 − 1)(q 1 + 1) − 6q 1 q 2 +q 1 q 2 +2q 1 +2q 2 = 2q 1 q 2 +3(q 1 +q 2 )−5 > 2q 1 q 2 ). Therefore, in order to prove that SD(n) > n for any number n of the form (2.9) with m ≥ 2, it is sufficient to prove that SD(n) > q sm m SD(n/q sm m ). But the last inequality is a consequence of equality (2.5). (2.10) We arrive at the following result. Recall that a positive integer m is called a perfect number if it is equal to the sum of all its proper divisors (i.e., of all divisors of m except of the number m itself). It is known (cf. [4,Theorem 5.10]) that every even perfect number is of the form 2 k−1 (2 k − 1), where the number 2 k − 1 is prime. Thus, our theorem implies the following result.
Corollary 2.6. The only perfect graceful number is 28.