Abstract
In this paper, we introduce a new type of labeling of a graph G with p vertices and q edges called edge δ− graceful labeling, for any positive integer δ, as a bijective mapping f of the edge set EG into the set δ,2δ,3δ,⋯,qδ such that the induced mapping f∗:VG→0,δ,2δ,3δ,⋯,qδ−δ, given by f∗u=∑uv∈EGfuvmodδk, where k=maxp,q, is an injective function. We prove the existence of an edge δ− graceful labeling, for any positive integer δ, for some cycle-related graphs like the wheel graph, alternate triangular cycle, double wheel graph Wn,n, the prism graph Πn, the prism of the wheel PWn, the gear graph Gn, the closed helm CHn, the butterfly graph Bn, and the friendship Frn.
Highlights
The graphs considered here will be finite, undirected, and simple where VðGÞ and EðGÞ will denote the vertex set and edge set of a graph G, respectively, p = jVðGÞj and q = jEðGÞj.A labeling of a graph is a mapping that carries graph elements to positive integers, subject to certain constraints
We introduce a new type of labeling of a graph G with p vertices and q edges called edge δ− graceful labeling, for any positive integer δ, as a bijective mapping f of the edge set EðGÞ into the set fδ, 2δ, 3δ,⋯,qδg such that the induced mapping f ∗ : VðGÞ → f0, δ, 2δ, 3δ,⋯,qδ − δg, given by f ∗ ðuÞ = ð∑uv∈EðGÞ f ðuvÞÞ mod ðδkÞ, where k = max ðp, qÞ, is an injective function
We prove the existence of an edge δ− graceful labeling, for any positive integer δ, for some cycle-related graphs like the wheel graph, alternate triangular cycle, double wheel graph W n,n, the prism graph Πn, the prism of the wheel PðW n Þ, the gear graph Gn, the closed helm CHn, the butterfly graph Bn, and the friendship Frn
Summary
Received 26 September 2019; Revised 13 November 2019; Accepted 22 November 2019; Published 20 January 2020. We introduce a new type of labeling of a graph G with p vertices and q edges called edge δ− graceful labeling, for any positive integer δ, as a bijective mapping f of the edge set EðGÞ into the set fδ, 2δ, 3δ,⋯,qδg such that the induced mapping f ∗ : VðGÞ → f0, δ, 2δ, 3δ,⋯,qδ − δg, given by f ∗ ðuÞ = ð∑uv∈EðGÞ f ðuvÞÞ mod ðδkÞ, where k = max ðp, qÞ, is an injective function. We prove the existence of an edge δ− graceful labeling, for any positive integer δ, for some cycle-related graphs like the wheel graph, alternate triangular cycle, double wheel graph W n,n , the prism graph Πn , the prism of the wheel PðW n Þ, the gear graph Gn , the closed helm CHn, the butterfly graph Bn , and the friendship Frn
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