A survey and a new class of graceful unicylic graphs
Max Pambe Biatch’, Jay S. Bagga, S. Arumugam · AKCE International Journal of Graphs and Combinatorics · 2020
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 , … , | E ( G ) | } such that when an edge xy is assigned the label | f ( x ) − f ( y ) | , the resulting set of edge labels is { 1 , 2 , … , | E ( G ) | } . When such a labeling exists, G is called graceful. Rosa showed that a cycle Cn ( n ≥ 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.