A Class of ‘n’ Distant Graceful Trees

Graceful graphs were first studied by Rosa in 1966. The Kotzig-Ringel graceful tree conjecture states that every tree has a graceful labeling. Aldred and McKay and others have used computer programs to show that trees of order up to 35 are graceful. Bagga et al. investigated algorithms for generating all graceful labelings of certain known classes of graceful graphs, including paths, cycles, and certain other classes of unicyclic graphs. The data generated by such algorithms has led to the discovery of new properties of such graceful labelings. In this paper we present a survey of graceful graph labeling algorithms and related complexity issues.DOI : http://dx.doi.org/10.22342/jims.0.0.14.1-9

Graceful Labeling of Balanced Stars of Paths

0-Centred and 0-ubiquitously graceful trees

<abstract><p>Graph labeling is a source of valuable mathematical models for an extensive range of applications in technologies (communication networks, cryptography, astronomy, data security, various coding theory problems). An edge $ \; \delta - $ graceful labeling of a graph $ G $ with $ p\; $ vertices and $ q\; $ edges, for any positive integer $ \; \delta $, is a bijective $ \; f\; $ from the set of edge $ \; E(G)\; $ to the set of positive integers $ \; \{ \delta, \; 2 \delta, \; 3 \delta, \; \cdots\; , \; q\delta\; \} $ such that all the vertex labels $ \; f^{\ast} [V(G)] $, given by: $ f^{\ast}(u) = (\sum\nolimits_{uv \in E(G)} f(uv)\; )\; mod\; (\delta \; k) $, where $ k = max (p, q) $, are pairwise distinct. In this paper, we show the existence of an edge $ \; \delta- $ graceful labeling, for any positive integer $ \; \delta $, for the following graphs: the splitting graphs of the cycle, fan, and crown, the shadow graphs of the path, cycle, and fan graph, the middle graphs and the total graphs of the path, cycle, and crown. Finally, we display the existence of an edge $ \; \delta- $ graceful labeling, for the twig and snail graphs.</p></abstract>

A substitution theorem for graceful trees and its applications

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.

A labeling of a graph is a mapping that carries some set of graph elements into numbers (usually positive integers). Graceful labeling of a graph with \(q\) edges is an injection from the set of its vertices to the sequence \(\lbrace 0,1,2, \ldots, q\rbrace\) such that the values of edges are all integers from \(1\) to \(q\), the value of an edge being absolute value of the difference between the integers attributed to its end vertices. In 2009, Sethuraman [8] posed a problem of labeling of arbitrary supersubdivision of graph. Fact that all cycles are not graceful in general, it was intresting to study same for arbitrary supersubdivisions of cycles. This inspiration lead us to some good results. In this paper, we prove that Even arbitrary supersubdivision of cycle is graceful. We also prove that Odd arbitrary supersubdivision of cycle \(C_n\), for \(n\) even is graceful and for \(n\) odd is not graceful.

A graceful labeling of a graph with q edges is a labeling of its vertices using the integers in [0, q], such that no two vertices are assigned the same label and each edge is uniquely identified by the absolute difference between the labels of its endpoints. The well known Graceful Tree Conjecture (GTC) states that all trees are graceful, and it remains open. It was proved in 1999 by Broersma and Hoede that there is an equivalent conjecture for GTC stating that all trees containing a perfect matching are strongly graceful (graceful with an extra condition). In this paper we extend the above result by showing that there exist infinitely many equivalent versions of the GTC. Moreover we verify these infinitely many equivalent conjectures of GTC for trees of diameter at most 7. Among others we are also able to identify new graceful trees and in particular generalize the ?-construction of Stanton-Zarnke (and later Koh- Rogers-Tan) for building graceful trees through two smaller given graceful trees.

A function f is called a graceful labelling of a graph G with q edges if f is an injection from the vertices of G to the set {0, 1, 2, …, q} such that, when each edge xy is assigned the label ∣f (x) − f (y)∣, the resulting edge labels are distinct. A graph G is said to be one modulo N graceful (where N is a positive integer) if there is a function ɸ from the vertex set of G to {0,1, N, (N + 1), 2N, (2N + 1), …, N (q − 1), N (q – 1) + 1} in such a way that (i) ɸ is 1 − 1 (ii) ɸ induces a bijection ɸ* from the edge set of G to {1, N + 1, 2N + 1, …, N (q − 1) + 1} where ɸ*(uv) = ∣ɸ(u) − ɸ(v)∣. In this paper we prove that the Supersubdivision of Ladder is one modulo N graceful for all positive integers N.

2-Role assignments on triangulated graphs

The spectral radius of trees on k pendant vertices

For positive integers m , n , K m , n represents the complete bipartite graph. We name the graph G = K m , n ⊙ K 2 as triangular extension of complete bipartite graph K m , n , since there is a triangle hanging from every vertex of K m , n . In this paper we show that G is graceful when m = n = 2l , for any integer l .

A connected graph G = (V, E) of order atleast two, with order p and size q is called vertex-graceful if there exists a bijection f : V → {1, 2, 3, ··· p} such that the induced function f ∗ : E → {0, 1, 2, ··· q − 1} defined by f ∗ (uv) = (f(u) + f(v))(mod q) is a bijection. The bijection f is called a vertex-graceful labeling of G. A subset S of the set of natural numbers N is called consecutive if S consists of consecutive integers. For any set X, a mapping f : X → N is said to be consecutive if f(X) is consecutive. A vertex-graceful labeling f is said to be strong if the function f1 : E → N defined by f1(e) = f(u)+ f(v) for all edges e = uv in E forms a consecutive set. It is proved that one vertex union of odd number of copies of isomorphic caterpillars is vertex-graceful and any caterpillar is strong vertex-graceful. It is proved that a spider with even number of legs (paths) of equal length appended to each vertex of an odd cycle is vertex-graceful. It is also proved that the graph lA(mj , n) is vertex-graceful for both n and l odd, 0 ≤ i ≤ n − 1, 1 ≤ j ≤ mi. Further, it is proved that the graph A(mj , n) is strong vertex-graceful for n odd, 0 ≤ i ≤ n − 1, 1 ≤ j ≤ mi.

Baskoro et al. [1] provided some constructions of new super edge-magic graphs from some old ones by attaching 1, 2, or 3 pendant vertices and edges. In this paper, we introduce (q, m)-super edge-magic total labeling and we give a construction of new super edge-magic graphs by attaching n pendant vertices and edges under some conditions, for any positive integer n. Also, we give a constraint on our construction.

A total k-labeling is a function fe from the edge set to first natural number ke and a function fv from the vertex set to non negative even number up to 2kv, where k=maxke,2kv. A vertex irregular reflexivek-labeling of a simple, undirected, and finite graph G is total k-labeling, if for every two different vertices x and x′ of G, wtx≠wtx′, where wtx=fvx+Σxy∈EGfexy. The minimum k for graph G which has a vertex irregular reflexive k-labeling is called the reflexive vertex strength of the graph G, denoted by rvsG. In this paper, we determined the exact value of the reflexive vertex strength of any graph with pendant vertex which is useful to analyse the reflexive vertex strength on sunlet graph, helm graph, subdivided star graph, and broom graph.