On d-graceful labelings

TL;DR

This paper introduces a generalized concept of graceful labelings called d-graceful labelings, explores their properties, and applies them to cyclic graph decompositions, providing complete solutions for paths and stars.

Contribution

It defines d-graceful and d-graceful α-labelings, investigates their existence in bipartite graphs, and applies these concepts to graph decompositions, extending classical labeling theories.

Findings

Complete solutions for paths and stars.

Partial results for even cycles and ladders.

New methods for cyclic graph decompositions.

Abstract

In this paper we introduce a generalization of the well known concept of a graceful labeling. Given a graph G with e=dm edges, we call d-graceful labeling of G an injective function from V(G) to the set {0,1,2,..., d(m+1)-1} such that {|f(x)-f(y)| | [x,y]\in E(G)} ={1,2,3,...,d(m+1)-1}-{m+1,2(m+1),...,(d-1)(m+1)}. In the case of d=1 and of d=e we find the classical notion of a graceful labeling and of an odd graceful labeling, respectively. Also, we call d-graceful \alpha-labeling of a bipartite graph G a d-graceful labeling of G with the property that its maximum value on one of the two bipartite sets does not reach its minimum value on the other one. We show that these new concepts allow to obtain certain cyclic graph decompositions. We investigate the existence of d-graceful \alpha-labelings for several classes of bipartite graphs, completely solving the problem for paths and stars and giving partial results about cycles of even length and ladders.