A substitution theorem for graceful trees and its applications

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.

0-Centred and 0-ubiquitously 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

Maximum gap labelings of graphs

Let T = (V, E) be a tree with a properly 2-colored vertex set. A bipartite labeling of T is a bijection φ: V → {1, …, |V|} for which there exists a k such that whenever φ(u) ≤ k < φ(v), then u and v have different colors. The α-size α(T) of the tree T is the maximum number of elements in the sets {|φ(u) − φ(v)|; uv ∈ E}, taken over all bipartite labelings φ of T. The quantity α(n) is defined as the minimum of α(T) over all trees with n vertices. In an earlier article (J Graph Theory 19 (1995), 201–215), A. Rosa and the second author proved that 5n/7 ≤ α(n) ≤ (5n + 4)/6 for all n ≥ 4; the upper bound is believed to be the asymptotically correct value of (n). In this article, we investigate the α-size of trees with maximum degree three. Let α3(n) be the smallest α-size among all trees with n vertices, each of degree at most three. We prove that α3(n) ≥ 5n/6 for all n ≥ 12, thus supporting the belief above. This result can be seen as an approximation toward the graceful tree conjecture—it shows that every tree on n ≥ 12 vertices and with maximum degree three has “gracesize” at least 5n/6. Using a computer search, we also establish that α3(n) ≥ n − 2 for all n ≤ 17. © 1999 John Wiley & Sons, Inc. J Graph Theory 31:7–15, 1999

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.

In this paper we show that a tree T with the following properties have graceful labeling. 1) T has a path H such that every pendant vertex of T has distance n (a fixed positive integer) from H . 2) Every vertex of T excluding one end vertex of H has even degree. Key words : Graceful labeling; n distant tree; Component moving transformation; Transfer of the first type

Any class of trees with n vertices is said to be a graceful graph if it has a graceful labeling. A graceful labeling of a graph G is a mapping of function f:V → {1,2,…, n} for every vertex such that each edge is labeled a distinct integer {1,2,…, n −1}, where the label of each edge is equal to the absolute value of the difference between a pair of vertices on its endpoints. In this paper, a symmetrical bi-Seesaw caterpillar is introduced and also proved that it can be gracefully labelled.

The binomial tree B0 consists of a single vertex. The binomial tree Bk is an ordered tree defined recursively. The binomial tree Bk consists of two binomial trees Bk−1 that are linked together: the root of one is the leftmost child of the root of the other. The popular Graceful Tree Conjecture states that every tree is graceful. In this paper, we show that binomial trees Bk is graceful for every k≥0.