Graceful Labellings Research Articles

Characterizations of kites as graceful graphs

We introduce and study an infinite family of graceful graphs, which we call kites. The kites are graphs where a path is joined with a graph "forming" a kite. We study and characterize three classes of the kites: kites formed by cycles known to be graceful, fan kites and lantern kites. Beside showing in a transparent way that all these graphs are graceful, we provide characterizations of these graphs among all simple graphs via three tools: via Sheppard's labelling sequences introduced in the 1970s and via labelling relations and graph chessboards. The latter are relatively new tools for the study of graceful graphs introduced by Haviar and Iva\v ska in 2015. The labelling relations are closely related to Sheppard's labelling sequences while the graph chessboards provide a~nice visualization of the graceful labellings.

Graceful Labellings of Variable Windmills Using Skolem-type Sequences

In this paper, we introduce graceful and near graceful labellings of several families of windmills. In particular, we use Skolem-type sequences to prove (near) graceful labellings exist for windmills with \(C_{3}\) and \(C_{4}\) vanes, and infinite families of \(3,5\)-windmills and \(3,6\)-windmills. Furthermore, we offer a new solution showing that the graph obtained from the union of \(t\) 5-cycles with one vertex in common (\(C_{5}^{t}\)) is graceful if and only if \(t \equiv 0, 3 \pmod{4}\) and near graceful when \(t \equiv 1, 2 \pmod{4}\).

Apportionable matrices and gracefully labelled graphs

To apportion a complex matrix means to apply a similarity so that all entries of the resulting matrix have the same magnitude. We initiate the study of apportionment, both by unitary matrix similarity and general matrix similarity. There are connections between apportionment and classical graph decomposition problems, including graceful labellings of graphs, Hadamard matrices, and equiangular lines, and potential applications to instantaneous uniform mixing in quantum walks. The connection between apportionment and graceful labellings allows the construction of apportionable matrices from trees. A generalization of the well-known Eigenvalue Interlacing Inequalities using graceful labellings is also presented. It is shown that every rank one matrix can be apportioned by a unitary similarity, but there are 2×2 matrices that cannot be apportioned. A necessary condition for a matrix to be apportioned by unitary matrix is established. This condition is used to construct a set of matrices with nonzero Lebesgue measure that are not apportionable by a unitary matrix.

A new class of graceful graphs: k-enriched fan graphs and their characterisations

The Graceful Tree Conjecture stated by Rosa in the mid 1960s says that every tree can be gracefully labelled. It is one of the best known open problems in Graph Theory. The conjecture has caused a great interest in the study of gracefulness of simple graphs and has led to many new contributions to the list of graceful graphs. However, it has to be acknowledged that not much is known about the structure of graceful graphs after 55 years. Our paper adds an infinite family of classes of graceful graphs to the list of known simple graceful graphs. We introduce classes of \(k\)-enriched fan graphs \(kF_n\) for all integers \(k, n\ge 2\) and we prove that these graphs are graceful. Moreover, we provide characterizations of the \(k\)-enriched fan graphs \(kF_n\) among all simple graphs via Sheppard's labelling sequences introduced in the 1970s, as well as via labelling relations and graph chessboards. These last approaches are new tools for the study of graceful graphs introduced by Haviar and Ivaska in 2015. The labelling relations are closely related to Sheppard's labelling sequences while the graph chessboards provide a nice visualization of graceful labellings. We close our paper with an open problem concerning another infinite family of extended fan graphs.

FIBONACCI AND SUPER FIBONACCI GRACEFUL LABELLINGS OF SOME TYPES OF GRAPHS

We consider the basic theoretical information regarding the Fibonacci graceful graphs. An injective function is said a Fibonacci graceful labelling of a graph of a size , if it induces a bijective function on the set of edges , where by the rule , for any adjacent vertices A graph that allows such labelling is called Fibonacci graceful. In this paper, we introduce the concept of super Fibonacci graceful labelling, narrowing the set of vertex labels, i.e. Four types of problems to be studied are selected. In the problem of the first type, the following question is raised: is there a graph that allows a certain kind of labelling, and under what conditions does this take place? The problem of the second type is the problem of construction: it is necessary, for a given system of requirements for the graph, to construct (at least one) its labelling that would satisfy this system. The following two types of problems relate to enumeration problems: for a given graph, determine the number of different Fibonacci and / or super Fibonacci graceful labellings; build all the different labellings of a given kind. As a result of solving these problems, functions were found that generate Fibonacci and super Fibonacci graceful labellings for graphs of cyclic structure; necessary and sufficient conditions for the existence of Fibonacci graceful labelling for disjunctive union of cycles, super Fibonacci graceful labelling for cycles, Eulerian graphs are obtained; the number of non-equivalent labellings of the cycle is determined; conditions for the existence of a super Fibonacci graceful labelling of a one-point connection of arbitrary connected super Fibonacci graceful graphs … …, are presented

On [formula omitted]-labellings of lobsters and trees with a perfect matching

A graceful labelling of a tree T is an injective function f:V(T)→{0,…,|E(T)|} such that {|f(u)−f(v)|:uv∈E(T)}={1,…,|E(T)|}. An α-labelling of a tree T is a graceful labelling f with the additional property that there exists an integer k∈{0,…,|E(T)|} such that, for each edge uv∈E(T), either f(u)≤k<f(v) or f(v)≤k<f(u). In this work, we prove that the following families of trees with maximum degree three have α-labellings: lobsters with maximum degree three, without Y-legs and with at most one forbidden ending; trees T with a perfect matching M such that the contraction T∕M has a balanced bipartition and an α-labelling; and trees with a perfect matching such that their contree is a caterpillar with a balanced bipartition. These results are a step towards the conjecture posed by Bermond in 1979 that all lobsters have graceful labellings and also reinforce a conjecture posed by Brankovic, Murch, Pond and Rosa in 2005, which says that every tree with maximum degree three and a perfect matching has an α-labelling.

Exponentially many genus embeddings of the complete graph K 12s+3

In this paper, we consider the problem of construction of exponentially many distinct genus embeddings of complete graphs. There are three approaches to solve the problem. The first approach is to construct exponentially many current graphs by the theory of graceful labellings of paths; the second approach is to find a current assignment of the current graph by the theory of current graph; the third approach is to find exponentially many embedding (or rotation) scheme of complete graph by finding exponentially many distinct maximum genus embeddings of the current graph. According to these three approaches, we can construct exponentially many distinct genus embeddings of complete graph K12s+3, which show that there are at least \(\frac{1}{2} \times {\left( {\frac{{200}}{9}} \right)^s}\) distinct genus embeddings for K12s+3.

Labelling Properties of Special Models Related with Networks

In the past few years many scientists have tried to develop models of networks and to investigate the mechanisms that determine the topology of complex networks. We define a class of graphs called (k,m)-dragon graphs and uniformly (k,m)-dragon graphs as referenced models of complex networks. These dragon graphs have some properties such as its admits (k,d)-odd-graceful labeling, graceful labellings, odd-graceful labellings and so on. In this paper, we proposed and defined the notion of some labelings of dragon graphs, the odd-graceful, graceful and total labellings of edges magic of these dragon graphs have been investigated in detail, and our methods can be algorithmic. Through the definition of dragon graphs and proofs of Theorems, we get some simple but usefulconclusions.

Decompositions into generalized Petersen graphs via graceful labelings

Abstract We prove that the generalized Petersen graph P 8 n , 3 admits an α-labeling for any integer n ⩾ 1 confirming that the conjecture posed by A. Vietri in [Vietri, A., Graceful labellings for an infinite class of generalised Petersen graphs, Ars Combin. 81 (2006), 247–255.] is true. We present also a result about the existence of d-divisible α-labelings of P 8 n , 3 . In such a way we obtain cyclic decompositions of the complete graph and of the complete multipartite graph into copies of P 8 n , 3 .

Graceful Labelling: State of the Art, Applications and Future Directions

This paper takes a close look at graceful labelling and its applications. We pay special attention to the famous Graceful Tree Conjecture, which has attracted a lot of interest and engaged many researchers over the last 40+ years, and yet to this day remains unsolved. We describe applications of graceful and graceful-like labellings of trees to several well known combinatorial problems and we expose yet another one, namely the connection between α-labelling of paths and near transversals in Latin squares. Finally, we show how spectral graph theory can be used to further the progress on the Graceful Tree Conjecture.

Complete triangulations of a given order generated from a multitude of nonisomorphic cubic graphs by current assignments

AbstractIt is known that for all sufficiently large s, there are at least \documentclass{article}\footskip=0pc\pagestyle{empty}\begin{document}$(\frac{5}{3})$\end{document}2s nonequivalent graceful labellings of the path on 2s + 1 vertices. Using this result, we construct exponentially many index one current graphs with current group ℤ12s + 7 such that many of the current graphs have different underlying graphs. The constructed current graphs for all sufficiently large sgenerate at least 30s nonisomorphic triangular embeddings of K12s + 7. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 324–334, 2009

From graceful labellings of paths to cyclic solutions of the Oberwolfach problem

Constraints are given on graceful labellings for paths that allow them to be used to construct cyclic solutions to the Oberwolfach Problem. We give examples of graceful labellings meeting these constraints and hence, infinitely many new families of cyclic solutions can be constructed.

Rook polynomials on two-dimensional surfaces and graceful labellings of graphs

By a simple trick we may generalise the rook polynomial for an n × n chessboard to various two-dimensional surfaces, the conventional chessboard corresponding to the torus. In the case of the Möbius band and the Klein bottle, there is a close connection to graceful labellings of graphs.

Graceful labellings of paths

It is easily shown that every path has a graceful labelling, however, in this paper we show that given almost any path P with n vertices then for every vertex v ∈ V ( P ) and for every integer i ∈ { 0 , … , n - 1 } there is a graceful labelling of P such that v has label i . We show precisely when these labellings can also be α -labellings. We then extend this result to strong edge-magic labellings. In obtaining these results we make heavy use of π -representations of α -labellings and review some relevant results of Kotzig and Rosa.

Triangular embeddings of complete graphs from graceful labellings of paths

We show that to each graceful labelling of a path on 2 s + 1 vertices, s ⩾ 2 , there corresponds a current assignment on a 3-valent graph which generates at least 2 2 s cyclic oriented triangular embeddings of a complete graph on 12 s + 7 vertices. We also show that in this correspondence, two distinct graceful labellings never give isomorphic oriented embeddings. Since the number of graceful labellings of paths on 2 s + 1 vertices grows asymptotically at least as fast as ( 5 / 3 ) 2 s , this method gives at least 11 s distinct cyclic oriented triangular embedding of a complete graph of order 12 s + 7 for all sufficiently large s.

Some Cyclic Solutions to the Three Table Oberwolfach Problem

We use graceful labellings of paths to give a new way of constructing terraces for cyclic groups. These terraces are then used to find cyclic solutions to the three table Oberwolfach problem, ${\rm OP}(r,r,s)$, where two of the tables have equal size. In particular we show that, for every odd $r \geq 3$ and even $r$ with $4 \leq r \leq 16$, there is a number $N_r$ such that there is a cyclic solution to ${\rm OP}(r,r,s)$ whenever $s \geq N_r$. The terraces we are able to construct also prove a conjecture of Anderson: For all $m \geq 3$, there is a terrace of ${\Bbb Z}_{2m}$ which begins $0, 2k, k, \ldots$ for some $k$.

A note on the number of graceful labellings of paths

With the help of a simple recursive construction we give a computer-assisted proof that the number of graceful labellings of a path of length n grows asymptotically at least as fast as (5/3) n . Results of this type have found surprising applications in topological graph theory.

Relaxed Graceful Labellings of Trees

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 $\{0,1,\dots,m\}$ such that the induced edge labelling $g$ defined by $g(uv)=|f(u)-f(v)|$ is a bijection to $\{1,\dots,m\}$. We here consider some relaxations of these conditions as applied to tree labellings: 1. Edge-relaxed graceful labellings, in which repeated edge labels are allowed, 2. Range-relaxed graceful labellings, in which the upper bound $m'$ is allowed to go higher than the number of edges, and 3. Vertex-relaxed graceful labellings, in which repeated vertex labels are allowed. The first of these had been looked at by Rosa and Širáň (1995). Here some linear bounds in the relevant metrics are given for range-relaxed and vertex-relaxed graceful labellings.

Ordered graceful labellings of the 2-star

A variation on the well-known graceful labelling problem for graphs is that of finding an ordered graceful labelling. A graceful labelling f :V(T)→{1,2,…,n} of a tree T of order n is called ordered graceful if, when the edges of the tree are oriented from the endvertex with larger label to the endvertex with smaller label, then every vertex has either indegree 0 or outdegree 0. In this study, we consider a technique for obtaining ordered graceful labellings of a 2-star T of order n and central vertex r, motivated by a conjecture of Cahit (Bull. ICA 12 (1994) 15–18) that just two non-equivalent ordered graceful labellings f of T are possible when n=6 s+1 or 6 s+3 with f( r)=1, and just two are possible when n=6 s+3 or 6 s+5 with f( r)=2. We show this conjecture is false when s>1 by constructing 2 s non-equivalent ordered graceful labellings in each of these cases. We use a link with Skolem sequences to aid some of the constructions.