On $d$-graceful Labelings

Abstract

We introduce a generalization of the well-known concept of graceful labeling. Given a graph $\mathrm{\Gamma }$ with $e=d.m$ edges, we define a $d$-graceful labeling of $G$ as an injective function $f:V(G)\to \{0,1,2,\dots ,d(m+1)–1\}$ such that $\{|f(x)–f(y)|:\{x,y\}\in E(\mathrm{\Gamma })\}$ = $\{1,2,3,\dots ,d(m+1)–1\}–\{m+1,2(m+1),\dots ,(d-1)(m+1)\}.$ 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 $\mathrm{\Gamma }$ a $d$-graceful labeling of $\mathrm{\Gamma }$ 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.