Abstract In this paper we define some new labellings for trees, called the in-improper and out-improper odd-graceful labellings such that some trees labelled with the new labellings can induce graceful graphs having at least a cycle. We, next, apply the new labellings to construct large scale of graphs having improper graceful/odd-graceful labellings or having graceful/odd-graceful labellings.

A graph G on m edges is considered graceful if there is a labelling f of the vertices of G with distinct integers in the set {0, 1, . . . ,m} such that the induced edge labelling g defined by g(uv) = |f(u) − f(v)| is a bijection to {1, . . . ,m}. We here consider some relaxations of these conditions as applied to tree labellings: 1. Edge-relaxed graceful labellings, in which repeated edge label...

there are many long-standing conjectures related with some labellings of trees. it is important to connect labellings that are related with conjectures. we find some connections between known labellings of simple graphs.

By strengthening an edge-decomposition technique for gracefully labelling a generalised Petersen graph, we provide graceful labellings for a new infinite family of such graphs. The method seems flexible enough to provide graceful labellings for many other classes of graphs in the future.

We show that to each graceful labelling of a path on 2s + 1 vertices, s ≥ 2, there corresponds a current assignment on a 3-valent graph which generates at least 22s cyclic oriented triangular embeddings of a complete graph on 12s + 7 vertices. We also show that in this correspondence, two distinct graceful labellings never give isomorphic oriented embeddings. Since the number of graceful labell...

We establish that all trees on at most 27 vertices admit graceful labellings and all trees on at most 26 vertices admit harmonious labellings. A graceful labelling of a graph G with q edges is an injection f : V (G) → {0, 1, 2, . . . , q} such that when each edge xy ∈ E(G) is assigned the label, |f(x) − f(y)|, all of the edge labels are distinct. A graph which admits a graceful labelling is sai...

A tree of order n is said to be graceful if the vertices can be assigned the labels {0, . . . , n−1} such that the absolute value of the differences in vertex labels between adjacent vertices generate the set {1, . . . , n− 1}. The Graceful Tree Conjecture is the unproven claim that all trees are graceful. We present major results known on graceful trees from those dating from the problem’s ori...