tandfonline.com – A survey and a new class of graceful unicylic graphs

Abstract

A graph G admits a graceful labeling if there is a one-to-one map f from the set of vertices of G to $\{0,1,2,\dots ,|E(G)\left|\right\}$ such that when an edge xy is assigned the label $|f(x)-f(y)|,$ the resulting set of edge labels is $\{1,2,\dots ,|E(G)\left|\right\}.$ When such a labeling exists, G is called graceful. Rosa showed that a cycle C_n ($n\ge 3$) is graceful if and only if n is congruent to 0 or 3 modulo 4. Truszczyński conjectured that unicyclic graphs, except the cycles in Rosa’s result are graceful. Several classes of unicyclic graphs are known to be graceful. In this article, we present a survey of results related to Truszczyński’s conjecture. We also present a new class of graceful unicyclic graphs.