# H-Coverings of Path-Amalgamated Ladders and Fans

Authors: Yijun Xiong; Huajun Wang; Muhammad Awais Umar; Yu-Ming Chu; Basharat Rehman Ali; Maria Naseem
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

LetG be a connected, simple graph with finite vertices v and edges e. A family G1,G2,…,Gp⊂G of subgraphs such that for all e∈E, e∈Gl, for some l,l=1,2,…,p is an edge-covering of G. If Gl≅ℍ, ∀l, then G has an ℍ-covering. Graph G with ℍ-covering is an ad,d-ℍ-antimagic if ψ:VG∪EG⟶1,2,…,v+e a bijection exists and the sum over all vertex-weights and edge-weights of ℍ forms a set ad,ad+d,…,ad+p−1d. The labeling ψ is super for ψVG=1,2,3,…,v and graph G is ℍ-supermagic for d=0. This manuscript proves results about super ℍ-antimagic labeling of path amalgamation of ladders and fans for several differences.


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## Body

## 1. Introduction and Preliminaries

LetG=VG,EG be a connected, finite, and simple graph [1, 2]. An edge-covering of G is a family G1,G2,…,Gp⊂G such that for all e∈EG, e∈EGl, for some l,l=1,2,…,p. If Gl≅ℍ, ∀l, then G has a ℍ-covering. Graph G with ℍ-covering is an ad,d-ℍ-antimagic if ∃ a one-to-one correspondence ψ:VG∪EG⟶1,2,…,v+e,∋forallℍ⊂G,ℍ≅G,(1)wtψℍ=∑ψvℍ+ψEℍ=ad,ad+d,…,ad+q−1d.ForψVG=1,2,3,…,VG, the labeling ψ becomes super ad,d-ℍ-antimagic and it would be ℍ-supermagic for d=0. Gutiérrez and Lladó, in [3], defined the ℍ-supermagic graphs for K1,m, Kp,m, Pm, and Cm for some subgraph ℍ. Jeyanthi and Selvagopal [4] proved ℍ-supermagic results for 2-connected graphs, k-polygonal snake, and one point union of m-disjoint paths. kCm-path, G≅K1,m+K1, book, and ladder-related graphs are ℍ-supermagic proved in [5]. Inayah et al. [6] introduced ℍ-antimagic graphs with weights forming an arithmetic progression. She also proved some bounds for ad and d for general graphs and fans. Susanto [7] derived bound for cycle-antimagic labeling of disjoint union of cycles. Recent results on ℍ-antimagic labeling of graphs can be seen in [8–14]. Also, in [12], Baca et al. discussed the tree-antimagicness of disconnected graphs. Recently, authors in [15] discussed the super a,1-tree-antimagicness of Sun graphs. The (super) ℍ-antimagic labeling is also related to a (super) d-antimagic labeling of type 1,1,0 of a plane graph [16]: a generalization of a face-magic labeling introduced by Lih [17]. Baca et al. proved d-antimagic labeling of type 1,1,1 for toroidal fullerenes in [18], while, in [19], Baca et al. proved labeling for plane graphs containing Hamiltonian paths. For more details, we refer [20–29] and the references therein. In the present article, we have studied super ℍ-antimagic labeling of path-amalgamation of ladder and fan for several differences, where ℍ is isomorphic to cycles C3, C4, C5, and P2-amalgamation of two cycles C3 and C4.


## 2. Path Amalgamation

In [30, 31], authors defined vertex amalgamation of m isomorphic copies of G, denoted with AmalG,u,m, for u∈G. Maryati et al. proved AmalG,v,m is G-supermagic in [32]. G-supermagic labeling of (i) edge amalgamation of 2-connected simple graph G is proved by Jeyanthi and Selvagopal in [4] and of (ii) AmalG,Pn,m by Salman and Maryati in [33]. In this paper, we extend this idea and use path graph Pm instead of a vertex to define a path amalgamation for two graphs ladders and fans. A ladder graph Lm is defined as the Cartesian product of Pm with P2, i.e., Lm≅Pm□P2. A fan graph Fm is defined as the join of Pm with an isolated vertex x, i.e., xFm≅Pm+x.Definition 1.
LetLm and Fm be ladder and fan graph for m≥3. A path amalgamation of ladder and fan denoted by AmalLm,Fm,Pm is obtained by identifying path Pm in ladder and fan.
The vertex set ofAmalLm,Fm,Pm is ai,bi,x:1≤i≤m and edge set is aibi,xbi:1≤i≤m∪aiai+1,bibi+1:1≤i≤m−1.
Figure1 depicts P6-amalgamation of ladder L6 and fan F6.Figure 1
P6-Amalgamation of ladder L6 and fan F6.Definition 2.
LetC3 and C4 be two cycles on 3 and 4 vertices, respectively. By identifying one edge of both cycles C3 and C4, we obtain P2-amalgamation denoted by AmalC4,C3,P2.
Figure2 depicts P2-amalgamation of cycles C4 and C3.
In Section3, we will study the existence of super ad,d-G′-antimagic labeling of path amalgamation of ladder and fan, where G′ is C3, C4, and AmalC3,C4,P2 for several differences.Figure 2
P2-Amalgamation of C3 and C4.


## 3. Superℍ-Antimagic Labeling of Path Amalgamation

From Figures1 and 2, it is clear that path amalgamation AmalLm,Fm,Pm has G′-coverings by subgraphs Gi,i=1,2,3,4, where G1:=C3, G2:=C4, G3:=C5, and G4:=AmalC3,C4,P2.The set of vertices and edges in subgraphsGi,i=1,2,3,4,j=1,2,…,m−1 are(2)VG1j=x,bj,bj+1,VG2j=aj,aj+1,bj,bj+1,VG3,4j=x,aj,aj+1,bj,bj+1,EG1j=xbj,xbj+1,bjbj+1,EG2j=ajaj+1,bjbj+1,ajbj,aj+1bj+1,EG3j=EG1∪EG2bjbj+1,EG4j=EG1∪EG2.Letψ be the total labeling of AmalLm,Fm,Pm, then wtψGij,i=1,2,3,4,j=1,2,…,m would be(3)wtψG1j=ψx+ψbj+ψbj+1+ψxbj+ψxbj+1+ψbjbj+1,(4)wtψG2j=ψaj+ψaj+1+ψbj+ψbj+1+ψajbj+ψajaj+1+ψbjbj+1+ψaj+1bj+1,(5)wtψG3j=wtψG1j+wtψG2j−ψbj−ψbj+1−2ψbjbj+1,(6)wtψG4j=wtψG3j+ψbjbj+1.Theorem 1.
LetAmalLm,Fm,Pm be amalgamation of ladder Lm and fan Fm and m≥3 be a positive integer. Then, AmalLm,Fm,Pm posses a super ad,d-Gi-antimagic labeling for i=1,2 and d∈0,1,2,…,8.Proof.
Consider the total labelingψ as follows:(7)ψdx=2m+1,ψdai=m2+i2i=even,d=0,1,2,3,i+12i=odd,d=0,1,2,3,ii=1,2,…,m,d=4,2i−1i=1,2,…,m,d=5,6,7,8,ψdbi=m+m2+i2i=even,d=1,2,4,m+i+12i=odd,d=1,2,4,m+ii=1,2,…,m,d=0,3,2ii=1,2,…,m,d=5,6,7,8,ψdaiai+1=4m−ii=1,2,…,m−1,d=0,1,…,8,ψdxbi=5m−1+m2+i2i=even,d=1,6,7,5m−1+i+12i=odd,d=1,6,7,6m−m2−i2i=even,d=0,6m−i+12i=odd,d=0,5m−1+ii=1,2,…,m,d=2,3,4,5,8,ψdbibi+1=3m+1−ii=1,2,…,m−1,d=0,1,2,3,5,2m+1+ii=1,2,…,m−1,d=4,6,2m+ii=1,2,…,m−1,d=7,8,ψdaibi=4m−1+ii=1,2,…,m,d=2,3,4m−1+m2+i2i=even,d=1,4,4m−1+i+12i=odd,d=1,4,5m−m2−i2i=even,d=0,5,8,5m−i+12i=odd,d=0,5,8,5m−ii=1,2,…,m,d=6,7.
Since,ψdVAmalLm,Fm,Pm=1,2,…,2m+1, ψd is a super labeling for d=0,1,…,8 and ψdEAmalLm,Fm,Pm=2m+2,2m+3,…,6m−1. Therefore, ψd is a total labeling.
Using (3) and (4) and j=1,2,…,m−1,(8)wtψ0Gij=19m+2−m2i=1,19m+2i=2,wtψ1Gij=17m+2+2m2+ii=1,17m+2+3m2+ii=2,wtψ2Gij=17m+m2+2+2ii=1,17m+2m2+2+2ii=2,wtψ3Gij=17m+2+3ii=1,17m+m2+2+3ii=2,wtψ4Gij=16m+m2+2+4ii=1,16m+2m2+2+4ii=2,wtψ5Gij=15m+3+5ii=1,17m−m2+2+5ii=2,wtψ6Gij=14m+m2+3+6ii=1,16m+2+6ii=2,wtψ7Gij=14m+m2+2+7ii=1,16m+1+7ii=2,wtψ8Gij=14m+2+8ii=1,16m−m2+1+8ii=2.
All the above equations show an arithmetic progression with differencesd∈0,1,2,…,8, respectively. This completes the proof.Theorem 2.
LetAmalLm,Fm,Pm be amalgamation of ladder Lm and fan Fm and m≥3 be a positive integer. Then, AmalLm,Fm,Pm possesses a super a′,d′-G3-antimagic labeling for d′∈0,3,5,6,8 and a″,d″-G4-antimagic labeling for d″∈0,1,2,…,8.Proof.
The subgraphsGi,i=3,4 contain the subgraphs Gi,i=1,2; therefore, differences d′ depends upon differences d obtained in Theorem 1. We will use the labeling ψd from Theorem 1. Let ν denote a vertex element in Gi,i=3,4 and ω denote and edge element in Gi,i=3,4; then, the total labeling ψd′′ is defined as follows:(9)ψd′′ν=ψd″″ν=ψdν,ψd′′ω=ψd″″ν=ψdω.
Sinceψd is a super total labeling, therefore ψd′′ is super total. For differences d′∈0,3,5,6,8, the total labeling ψd′″ of G3j is obtained as follows:(10)ψ0′G3j=ψ0G3j,ψ3′G3j=ψ1G3j,ψ5′G3j=ψ2G3j,ψ6′G3j=ψ3G3j,ψ8′G3j=ψ5G3j.
For differencesd″∈1,2,5,6,7,8,10, the total labeling ψd′″ of G4j is obtained as follows:(11)ψ1″G4j=ψ0G4j,ψ2″G4j=ψ1G4j,ψ5″G4j=ψ3G4j,ψ6″G4j=ψ4G4j,ψ7″G4j=ψ5G4j,ψ8″G4j=ψ7G4j,ψ10″G4j=ψ8G4j.
Using (5) and (6) and j=1,2,…,m−1,(12)wtψd′′G3j=30m+1−m2d′=0,26m+4m2+1+3id′=3,26m+2m2+1+5id′=5,26m+2m2+1+6id′=6,26m−m2+1+8id′=8,wtψd″″G4j=33m+2−m2−id′=1,29m+4m2+3+2id′=2,29m+m2+2+5id′=5,28m+2m2+2+6id′=6,29m−m2+2+7id′=7,28m+m2+1+8id′=8,28m−m2+1+10id′=10.
All the above equations show an arithmetic progression with required differencesd′ and d″. This completes the proof.


## 4.ℍ-Antimagic Coverings for Disjoint Union

Theorem 3.
Let path amalgamation of ladder and fanAmalLm,Fm,Pm,m≥3 admit Gi-supermagic labeling for i=1,2,3,4. Then, the disjoint union of arbitrary number of copies of AmalLm,Fm,Pm, i.e., τAmalLm,Fm,Pm also admits a Gi-supermagic labeling for i=1,2,3 and super a,1-G4-antimagic for τ≥1 a positive integer.Proof.
Letτ≥1 be a positive integer. In our proof, uk,k=1,2,…,τ, denotes a vertex or an edge in the kth copy of the path amalgamation of ladder and fan, denoted by G:=AmalLm,Fm,Pmk, corresponding to u in AmalLm,Fm,Pm, i.e., u∈VAmalLm,Fm,Pm∪EAmalLm,Fm,Pm. In the same way, let Girk,i=1,2,3,4,k=1,2,…,τ,r=1,2,…,m−1, be the subgraph in the kth copy of AmalLm,Fm,Pm corresponding to the subgraph Gir,i=1,2,3,4,r=1,2,…,m−1 in AmalLm,Fm,Pm. The total labeling h′ of τAmalLm,Fm,Pm is defined as follows:(13)ϕuk=τϕu−1+ku∈VAmalLm,Fm,Pm,τϕu+1−ku∈EAmalLm,Fm,Pm.
First, we will prove that vertices ofG:=AmalLm,Fm,Pmk use integers from 1 up to pτ under the labeling ϕ, where p is number of vertices in graph AmalLm,Fm,Pm, i.e.,(14)ϕVG=1,τ+1,2τ+1,…,p−1τ+1,k=1,2,τ+2,2τ+2,…,p−1τ+2,k=2,⋯⋯⋯j,τ+j,2τ+j,…,p−1τ+j,k=j,⋯⋯⋯τ,2τ,3τ,…,pτ,k=τ.
Secondly, for edges of∪k=1τAmalLm,Fm,Pmk with EG=q, under the labeling ϕ,(15)ϕEG=p+1τ,p+2τ,p+3τ,…,p+qτk=1,p+1τ−1,p+2τ−1,p+3τ−1,…,p+qτ−1k=2,⋯⋯⋯p+1τ+1−j,p+2τ+1−j,p+3τ+1−j,…,p+qτ+1−jk=j,⋯⋯⋯pτ+1,p+1τ+1,p+2τ+1,…,p+q−1τ+1k=τ.
From equations (14) and (15), it is clear the labeling ϕ is a total labeling, since ϕ is a bijection between the integers 1,2,…,p+qτ and the vertices and edges of ∪k=1τAmalLm,Fm,Pmk.
Under the total labelingϕ, the Girk-weights, i=1,2,3,4,1≤k≤τ,1≤r≤m−1 would be(16)wtϕGirk=∑v∈VGirkϕv+∑e∈EGirkϕe=∑v∈VGirkτϕv−1+k+∑e∈EGirkτϕe+1−k=τ∑v∈VGirkϕv−τVGirk+kVGirk+τ∑e∈EGirkϕe+EGirk−kEGirk,where r is the number of Gik’s i=1,2,3,4,r=1,2,…,m−1 in AmalLm,Fm,Pmk:(17)wtϕGirk=τ∑v∈VGirkϕv+∑e∈EGirkϕe−τVGirk+EGirkkVGirk−kEGirk=τwtϕGirk−τVGirk+EGirk+kVGirk−kEGirk.
As everyGirk, i=1,2,3,4,k=1,2,…,τ,r=1,2,…,m−1, is isomorphic to the cycle Gi,i=1,2,3,4, it holds(18)VG1rk=VG1r=3,VG2rk=VG2r=4,VG3rk=VG3r=5,VG4rk=VG4r=5,EG1rk=EG1r=3,EG2rk=EG2r=4,EG3rk=EG3r=5,EG4rk=EG4r=6.
Thus, for theGi-weights for i=1,2,3,4, we obtain(19)wtϕG1rk=τwtϕG1rk+31−τ=τ229m−1+43+31−τ=τ229m−1+35+3,wtϕG2rk=τwtϕG2rk+41−τ=τ229m−1+43+31−τ=τ229m−1+35+3,wtϕG3rk=τwtϕG3rk+51−τ=τ229m−1+43+51−τ=τ229m−1+35+5,wtϕG4rk=τwtϕG1rk+61−τ=τ229m−1+43+61−τ−k=τ229m−1+35+6−k.
It is easy to see that the set of allGirk-weights for i=1,2,3 consists of the same integers and G4-weight consists of consecutive integers in ∪k=1τAmalLm,Fm,Pmk. Thus, the graph ∪k=1τAmalLm,Fm,Pm is a Gi,i=1,2,3-supermagic and super a,1-G4-antimagic. This completes the proof.



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