On \((a, b; n)\)-Graceful Labeling of Path \(P_n\)

Abstract

This paper devotes to solving the following conjecture proposed by Gvozdjak: “An \((a, b; n)\)-graceful labeling of \(P_n\) exists if and only if the integers \(a, b, n\) satisfy (1) \(b – a\) has the same parity as \(n(n + 1)/2\); (2) \(0 < |b – a| \leq (n + 1)/2\) and (3) \(n/2 \leq a + b \leq 3n/2\).'' Its solving can shed some new light on solving the famous Oberwolfach problem. It is shown that the conjecture is true for every \(n\) if the conjecture is true when \(n \leq 4a + 1\) and \(a\) is a fixed value. Moreover, we prove that the conjecture is true for \(a = 0, 1, 2, 3, 4, 5, 6\).