Related papers: Growing Graceful and Harmonious Trees
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…
Graceful tree conjecture is a well-known open problem in graph theory. Here we present a computational approach to this conjecture. An algorithm for finding graceful labelling for trees is proposed. With this algorithm, we show that every…
Four algorithms giving rise to graceful graphs from a known (non)graceful graph are described. Some necessary conditions for a graph to be highly graceful and critical are given. Finally some conjectures are made on graceful, critical and…
In an attempt to prove the Graceful Tree Conjecture, we present two propagation of graphs. The first is to propagate graceful graphs, and the second is to propagate trees from a gracefully labeled tree. The motivation in propagating such…
We prove via a composition lemma, the Kotzig-Ringel-Rosa conjecture, better known as the Graceful Labeling Conjecture. We also prove via a stronger version of the composition lemma a stronger form of the Graceful Labeling Conjecture.
Let T=(V,E) be a tree with vertex set V and edge set E. A graceful labelling f of T is an injective function f from V into {0, 1, ..., |E|} such that if edge uv is assigned the label g(uv)=|f(u)-f(v)| then the function g from E into {1,…
Given a graph $G$, a labeling of $G$ is an injective function $f:V(G)\rightarrow\mathbb{Z}_{\ge 0}$. Under the labeling $f$, the label of a vertex $v$ is $f(v)$, and the induced label of an edge $uv$ is $|f(u) - f(v)|$. The labeling $f$ is…
Graph labelling is one of the noticed contexts in combinatorics and graph theory. Graceful labelling for a graph $G$ with $e$ edges, is to label the vertices of $G$ with $0, 1, \cdots, e$ such that, if we specify to each edge the difference…
In this paper, we construct additively graceful signed graphs S from a given graph G that may be additively graceful or not be additively graceful. We also show the construction of additively graceful signed graphs from additively graceful…
A \emph{graceful labeling} of a graph $G$ is an injective function $f : V(G) \to \{0, \ldots, |E(G)|\}$ such that $\{\,|f(u)-f(v)| : uv \in E(G)\,\} = \{1, \ldots, |E(G)|\}$. If such a labeling exists, then we call $G$ \emph{graceful}.…
A graceful labelling of a graph G is an injective function f from the set of vertices of G into the set {0,1,...,|EG|} such that if edge uv is assigned the label |f(u)-f(v)| then all edge labels have distinct values. A strong graceful…
In graph theory, a graceful labeling of a graph with m edges is a labeling of its vertices with a subset of the integers ranging from 0 to m inclusive, such that no two vertices share a label, and each edge is uniquely identified by the…
Generating trees are a useful technique in the enumeration of various combinatorial objects, particularly restricted permutations. Quite often the generating tree for the set of permutations avoiding a set of patterns requires infinitely…
Graham and Sloane proposed in 1980 a conjecture stating that every tree has a harmonious labelling, a graph labelling closely related to additive base. Very limited results on this conjecture are known. In this paper, we proposed a…
The Graceful Tree Conjecture of Rosa from 1967 asserts that the vertices of each tree T of order n can be injectively labelled by using the numbers {1,2,...,n} in such a way that the absolute differences induced on the edges are pairwise…
We give a short and direct proof of a remarkable identity that arises in the enumeration of labeled trees with respect to their indegree sequence, where all edges are oriented from the vertex with lower label towards the vertex with higher…
The concept of graceful labels was proposed by Rosa, scholars began to study graceful labels of various graphs and obtained relevant results.Let the graph is a bipartite graceful graph, we have proved some graphs are graceful labeling in…
In this paper, we propose an algorithm to generate all possible graceful graphs (including trees) containing n vertices as lattice paths in a certain triangular lattice defined below. This lattice that corresponds to graphs containing n…
A function $f$ is a \textit{graceful labelling} of a graph $G=(V,E)$ with $m$ edges if $f$ is an injection $f:V\mapsto \{0,1,2,\dots,m\}$ such that each edge $uv \in E$ is assigned the label $|f(u)-f(v)|$, and no two edge labels are the…
Inspired by the network routing literature \cite{aggarwal1996efficient}, we develop what we call a ``Pre-Emptive Greedy Algorithm" to embed bounded degree induced trees in sparse expanders. This generalises a powerful and central result of…