Gracefully Cultivating Trees on a Cycle

Figueroa-Centeno, Ramón M.; Ichishima, Rikio; Muntaner-Batle, Francesc A.; Oshima, Atsushi

doi:10.1016/j.endm.2015.05.020

Cited by 3 publications

(4 citation statements)

References 6 publications

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“…The study of graceful labelings of unicyclic graphs has received some attention during the last decades, several papers can be found in the specialized literature. In [5], Ichishima et al, provided a pair of methods to build unicyclic graceful graphs. Basically, their methods can be described as the vertex amalgamation of α-labeled trees of the same size one per vertex of the cycle C n .…”

Section: Pendent Paths On α-Cyclesmentioning

confidence: 99%

“…The process is a small variation of the procedure described in Theorem 3.1. This technique differs from the ideas in [5] because the paths that we are attaching to the vertices of the cycle have different sizes. In Section 6, we study some conditions, that when are satisfied by an α-labeled graph, allow us to replace these paths by, what we called, equivalent α-trees, which increases substantially the number of known gracefully labeled unicyclic graphs.…”

Section: Pendent Paths On α-Cyclesmentioning

confidence: 99%

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Alpha graphs with different pendent paths

Barrientos¹

2020

EJGTA

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Graceful labelings are an effective tool to find cyclic decompositions of complete graphs and complete bipartite graphs. The strongest kind of graceful labeling, the α-labeling, is in the center of the research field of graph labelings, the existence of an α-labeling of a graph implies the existence of several, apparently non-related, other labelings for that graph. Furthermore, graphs with α-labelings can be combined to form new graphs that also admit this type of labeling. The standard way to combine these graphs is to identify every vertex of a base graph with a vertex of another graph. These methods have in common that all the graphs involved, except perhaps the base, have the same size. In this work, we do something different, we prove the existence of an α-labeling of a tree obtained by attaching paths of different lengths to the vertices of a base path, in such a way that the lengths of the pendent paths form an arithmetic sequence with difference one, where consecutive vertices of the base path are identified with paths which lengths are consecutive elements of the sequence. These α-trees are combined in several ways to generate new families of α-trees. We also prove that these trees can be used to create unicyclic graphs with an α-labeling. In addition, we show that the pendent paths can be substituted by equivalent α-trees to produce new α-trees, obtaining in this manner a quite robust category of α-trees.

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Section: Pendent Paths On α-Cyclesmentioning

confidence: 99%

Section: Pendent Paths On α-Cyclesmentioning

confidence: 99%

Alpha graphs with different pendent paths

Barrientos¹

2020

EJGTA

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“…The same year, Figueroa-Centeno et al [7] provided two construction methods that generate a graceful labeling of certain unicyclic graphs obtained by amalgamating some vertices of the cycle and some vertices of trees. The cycle has n vertices where n 0 or 3 (mod 4).…”

Section: The Conjecture Of Truszczy Nskimentioning

confidence: 99%

A survey and a new class of graceful unicylic graphs

Biatch

Bagga

Arumugam

2020

AKCE International Journal of Graphs and Combinatorics

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A graph G admits a graceful labeling if there is a one-to-one map f from the set of vertices of G to f0, 1, 2, :::, jEðGÞjg such that when an edge xy is assigned the label jf ðxÞ À f ðyÞj, the resulting set of edge labels is f1, 2, :::, jEðGÞjg: When such a labeling exists, G is called graceful. Rosa showed that a cycle C n (n ! 3) is graceful if and only if n is congruent to 0 or 3 modulo 4. Truszczy nski conjectured that unicyclic graphs, except the cycles in Rosa's result are graceful. Several classes of unicyclic graphs are known to be graceful. In this article, we present a survey of results related to Truszczy nski's conjecture. We also present a new class of graceful unicyclic graphs.

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“…• Figueroa-Centeno et al [9] have provided an interesting construction to form a graceful unicyclic graph from a set of α-labeled trees with some special property [A graceful labeling f of a graph G is called an α-labeling if there exists an integer λ such that, min{ f (x), f (y)} ≤ λ < max{ f (x), f (y)} for each edge (x, y) in G].…”

Section: Introductionmentioning

confidence: 99%

Generating graceful unicyclic graphs from a given forest

Sethuraman

Murugan

2020

AKCE International Journal of Graphs and Combinatorics

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Acharya (1982) proved that every connected graph can be embedded in a graceful graph. The generalization of this result that, any set of graphs can be packed into a graceful graph was proved by Sethuraman and Elumalai (2005). Recently, Sethuraman et al. ( 2016) have shown that, every tree can be embedded in an graceful tree. Inspired by these fundamental structural properties of graceful graphs, in this paper, we prove that any acyclic graph can be embedded in an unicyclic graceful graph. This result is proved algorithmically by constructing graceful unicyclic graphs from a given acyclic graph. Our result strongly supports the Truszczynski's Conjecture that "All unicyclic graphs except the cycle C n with n ≡ 1 or 2(mod 4) are graceful".

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Gracefully Cultivating Trees on a Cycle

Cited by 3 publications

References 6 publications

Alpha graphs with different pendent paths

Alpha graphs with different pendent paths

A survey and a new class of graceful unicylic graphs

Generating graceful unicyclic graphs from a given forest

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