Super (a, d)-H-antimagic labeling of subdivided graphs
-
Amir Taimur
,
Adeela Mumtaz
Abstract
A simple graph G = (V, E) admits an H-covering, if every edge in E(G) belongs to a subgraph of G isomorphic to H. A graph G admitting an H-covering is called an (a, d)-H-antimagic if there exists a bijective function f : V(G) ∪ E(G) → {1, 2, …, |V(G)| + |E(G)|} such that for all subgraphs H′ isomorphic to H the sums ∑v∈V(H′)f(v) + ∑e∈E(H′)f(e) form an arithmetic sequence {a, a + d, …, a + (t − 1)d}, where a > 0 and d ≥ 0 are integers and t is the number of all subgraphs of G isomorphic to H. Moreover, if the vertices are labeled with numbers 1, 2, …, |V(G)| the graph is called super. In this paper we deal with super cycle-antimagicness of subdivided graphs. We also prove that the subdivided wheel admits an (a, d)-cycle-antimagic labeling for some d.
1 Introduction
Let G = (V, E) be a finite simple graph with the vertex set V(G) and the edge set E(G). An edge-covering of G is a family of subgraphs H1, H2, …, Ht such that each edge of E belongs to at least one of the subgraphs Hi, i = 1, 2, …, t. Then it is said that G admits an (H1, H2, …, Ht)-(edge) covering. If every subgraph Hi is isomorphic to a given graph H, then the graph G admits an H-covering. A bijective function f : V(G) ∪ E(G) → {1,2, …, |V(G)| + |E(G)| } is an (a, d)-H-antimagic labeling of a graph G admitting an H-covering whenever, for all subgraphs H′ isomorphic to H, the H′-weights
form an arithmetic progression a, a + d, …, a + (t − 1)d, where a > 0 and d ≥ 0 are two integers, and t is the number of all subgraphs of G isomorphic to H. Such a labeling is called super if the smallest possible labels appear on the vertices. A graph that admits a (super) (a, d)-H-antimagic labeling is called (super) (a, d)-H-antimagic. For d = 0 it is called H-magic and H-supermagic, respectively.
The H-(super)magic labelings were first studied by Gutiérrez and Lladó [1] as an extension of the edge-magic and super edge-magic labelings introduced by Kotzig and Rosa [2] and Enomoto, Lladó, Nakamigawa and Ringel [3], respectively. In [1] are considered star-(super)magic and path-(super)magic labelings of some connected graphs and it is proved that the path Pn and the cycle Cn are Ph-supermagic for some h. Lladó and Moragas [4] studied the cycle-(super)magic behavior of several classes of connected graphs. They proved that wheels, windmills, books and prisms are Ch-magic for some h. Maryati, Salman, Baskoro, Ryan and Miller [5] and also Salman, Ngurah and Izzati [6] proved that certain families of trees are path-supermagic. Ngurah, Salman and Susilowati [7] proved that chains, wheels, triangles, ladders and grids are cycle-supermagic. Maryati, Salman and Baskoro [8] investigated the G-supermagicness of a disjoint union of c copies of a graph G and showed that the disjoint union of any paths is cPh-supermagic for some c and h.
The (a, d)-H-antimagic labeling was introduced by Inayah, Salman and Simanjuntak [9]. In [10] there are investigated the super (a, d)-H-antimagic labelings for some shackles of a connected graph H. In [11] was proved that wheels are cycle-antimagic. In [12] it was shoved that if a graph G admits a (super) (a, d)-H-antimagic labeling, where d = |E(H)| − |V(H)|, then the disjoint union of m copies of the graph G, denoted by mG, admits a (super) (b, d)-H-antimagic labeling as well. Rizvi, et al. [13] proved the disjoint union of isomorphic copies of fans, triangular ladders, ladders, wheels, and graphs obtained by joining a star K1,n with K1, and also disjoint union of non-isomorphic copies of ladders and fans are cycle-supermagic.
In this paper we will discuss a super cycle-atimagicness of subdivided graphs. We show that the property to be super (a, d)-H-antimagic is hereditary according to the operation of subdivision of edges. We prove that if a graph G is super cycle-antimagic then the subdivided graph S(G) also admits a super cycle-antimagic labeling. Moreover, we show that the subdivided wheel is super (a, d)-cycle-antimagic for wide range of differences.
2 Subdivided graphs
Let us consider the graph S(G) obtained by subdividing some edges of a graph G, thus by inserting some new vertices to the original graph G. Equivalently, the graph S(G) can by obtained from G by replacing some edges of G by paths. The topic of subdivided graphs has been widely studied in recent years, for example see [14].
Let G be a graph admitting H-covering given by t subgraphs H1, H2, …, Ht isomorphic to H. Let us consider the subgraphs SG(Hi), i = 1, 2, …, t, corresponding to Hi in S(G). If these subgraphs are all isomorphic to a graph, let us denote it by the symbol SG(H), then the graph S(G) admits SG(H)-covering.
The next theorem shows that the property of being super (a, d)-H-antimagic is hereditary according to the operation of subdivision of edges.
Theorem 2.1
LetGbe a super (a, d)-H-antimagic graph and letHi, i = 1, 2, …, t, be all subgraphs ofGisomorphic toH. IfSG(Hi), i = 1, 2, …, t, are all subgraphs ofS(G) isomorphic toSG(H) then the graphS(G) is a super (b, d)-S(H)-antimagic graph.
Proof
Let G be a super (a, d)-H-antimagic graph and let Hi, i = 1, 2, …, t, be all subgraphs of G isomorphic to H. Let f be a super (a, d)-H-antimagic labeling of G, thus f : V(G) ∪ E(G) → {1, 2, …, |V(G)| + |E(G)|} such that the vertices of G are labeled with numbers 1, 2, …, |V(G)| and the weights of subgraphs Hi, i = 1, 2, …, t,
form an arithmetic progression a, a + d, …, a + (t − 1)d, where a > 0 and d ≥ 0 are two integers, i.e.,
Let us consider the graph S(G) obtained from G by inserting p new vertices, say v1, v2, …, vp, to the edges of G. Let SG(Hi), i = 1, 2, …, t, be all subgraphs of S(G) isomorphic to SG(H). Then S(G) admits the SG(H)-covering. Let r denote the number of new vertices inserted to every subgraph SG(Hi), i = 1, 2, …, t.
We define a labeling g of S(G) in the following way
Evidently, the vertices of S(G) are labeled with distinct numbers 1, 2, …, |V(G)| + p.
Let us choose an orientation of edges in G. According to this orientation we orient the edges in S(G). To an arc uv in G there will correspond the oriented path Puv with initial vertex u and terminal vertex v in S(G). The arcs of S(G) we label such that
The edges are labeled with distinct numbers from the set |V(G)| + p + 1, |V(G)| + p + 2, …, |V(G)| + |E(G)| + 2p. Now we evaluate the weights of subgraphs SG(Hi), i = 1, 2, …, t, under the labeling g. Immediately using the structure of the subgraph SG(Hi) and the definition of the labeling g we get
As |E(Hi)| = |E(H)| for i = 1, 2, …, t we obtain that the weights of SG(Hi) depend on the weights of Hi which form an arithmetic sequence with a difference d, see (1). This implies that the set of weights SG(Hi) also forms an arithmetic sequences with the difference d and the initial term a + |E(H)|p + (2|V(G)| + |E(G)| + 2p + 1)r. This concludes the proof. □
Combining Theorem 2.1 with some results on (a, d)-cycle-antimagic graphs we immediately obtain new classes of graphs that are (b, d)-cycle-antimagic. Note, that it is not needed to consider only regular subdivisions of graphs.
3 Subdivided wheels
A wheel Wn is a graph obtained by joining a single vertex to all vertices of a cycle on n vertices. The vertex of degree n is called the central vertex, or the hub vertex, and the remaining vertices are called the rim vertices. The edges adjacent to the central vertex are called spokes and the remaining edges are called rim edges. Let us denote by the symbol Wn(r, s) the graph obtained by inserting r, r ≥ 0, new vertices to every rim edge and s, s ≥ 0, new vertices to every spoke in the wheel Wn. Note, that the graph isomorphic to subdivided wheel Wn(r, 0) is also known as the Jahangir graph Jn,r+1.
In [11] it was proved that wheels are cycle-antimagic.
Theorem 3.1
([11]). Letkandn ≥ 3 be positive integers. The wheelWnis super (a, 1)-Ck-antimagic for everyk = 3, 4, …, n − 1, n + 1.
Immediately using Theorem 2.1 we obtain that subdivided wheels admit cycle-antimagic labeling with difference 1.
Corollary 3.2
Letk, n ≥ 3, r ≥ 0, s ≥ 0 be integers. The subdivided wheelWn(r, s) is super (a, 1)-Ck+(k−2)r+2s-antimagic for everyk = 3, 4, …, n − 1, n + 1.
In the next theorem we will deal with the cycle-antimagicness of the subdivided wheel Wn(1, 1). We prove that this graph admits a super (a, d)-C6-antimagic labeling for d ∈ {0, 1, …,5}.
Theorem 3.3
The subdivided wheelWn(1, 1), n ≥ 3, is super (a, d)-C6-antimagic ford ∈ {0, 1, …,5}.
Proof
Let us denote the vertices and edges of Wn(1, 1) such that
where the indices are taken modulo n.
For d = 1 the result follows from Corollary 3.2. For d ∈ {0, 2,3, 4,5} we define a total labeling gd:V(Wn(1, 1)) ∪ E(Wn(1, 1)) → {1, 2, …, 7n + 1} in the following way.
We denote by the symbol
It is a simple mathematical exercise to prove that for every i, 1 ≤ i ≤ n, the 6-cycle-weights are:
Hence the weights of cycles C6 form an arithmetic sequence with differences d = 0, 2,3, 4,5, respectively. This concludes the proof. □
Combining Theorem 2.1 and Theorem 3.3 we immediately obtain the following result.
Theorem 3.4
The subdivided wheelWn(r, s), n ≥ 3, r ≥ 1 ands ≥ 1 is super (a, d)-Cr+2s+3-antimagic for d ∈ {0, 1, 2, 3, 4, 5}.
In the next section we will deal with the subdivided wheel Wn(r, 0), n ≥ 3, r ≥ 1. Let us denote the vertices and the edges of Wn(r, 0) such that
The subdivided wheel Wn(r, 0), n ≥ 3, r ≥ 1, has n vertices of degree 3, nr vertices of degree 2 and one vertex of degree n. The size of Wn(r, 0) is n(r + 2).
The subdivided wheel Wn(r, 0) admits the Cr+3-covering consisting of n cycles Cr+3. Let us denote these cycles by the symbols
The following theorem shows the existence of a super (a, d)-Cr+3-antimagic labeling for Wn(r, 0) for every odd difference form 1 up to 2r − 3.
Theorem 3.5
The subdivided wheelWn(r, 0), n ≥ 3, r ≥ 1, is super (a, d)-Cr+3-antimagic ford = 1 whenr = 1 and ford ≡ 1 (mod 2), 1 ≤ d ≤ 2r − 3 whenr ≥ 1.
Proof
For r = 1 the result follows from Corollary 3.2. Let r ≥ 2 and let d be an odd positive integer, 1 ≤ d ≤ 2r − 3. Let fd:V(Wn(r, 0)) ∪ E(Wn(r, 0)) → {1, 2, …, n(2r + 3) + 1} be a labeling of Wn(r, 0), n ≥ 3, r ≥ 2, defined in the following way.
It is easy to see that fd is a bijection as
Under the labeling fd the weights of cycles
Moreover, the weight of the cycle
This proves that fd is a super (a, d)-Cr+3-antimagic labeling of Wn(r, 0) for d ≡ 1 (mod 2), 1 ≤ d ≤ 2r − 3 and a = 2nr2 + 7nr + 6n + 3r + 9 − (d + 1)(n + 1)/2. □
In the next theorem we prove that the graph Wn(r, 0) admits super (a, d)-Cr+3-antimagic labelings also for even differences.
Theorem 3.6
The subdivided wheelWn(r, 0), r ≥ 1, is super (a, d)-Cr+3-antimagic ford = 0 whenr = 1, n ≥ 5 and ford ≡ 0 (mod 2), 0 ≤ d ≤ 2r − 4 whenr ≥ 2, n ≥ 3.
Proof
Lladó and Moragas [4] proved that the wheel Wn, n ≥ 5 odd, is (a, 0)-C3-antimagic. From Corollary 3.2 we obtain that Wn(r, 0), n ≥ 5, r ≥ 1, is super (b, 0)-Cr+3-antimagic.
Let r ≥ 2, n ≥ 3 be positive integers. Let d be an even integer, 0 ≤ d ≤ 2r − 4. Let fd:V(Wn(r, 0)) ∪ E(Wn(r, 0)) → {1, 2, …, n(2r + 3) + 1} be a labeling of Wn(r, 0), n ≥ 3, r ≥ 1, defined in the following way.
The labeling gd is a bijection. Under the labeling gd the weights of cycles
For the weight of the cycle
We showed that gd is a super (a, d)-Cr+3-antimagic labeling of Wn(r, 0) for d ≡ 0 (mod 2), 0 ≤ d ≤ 2r − 4 and a = 2nr2+8nr + 8n + 3r + 8 − dn/2 + d/2. □
Combining Theorem 3.5 and Theorem 3.6 we immediately obtain that the subdivided wheel Wn(r, 0), n ≥ 5, is cycle-antimagic for wide range of differences.
Theorem 3.7
The subdivided wheelWn(r, 0), n ≥ 5, is super (a, d)-Cr+3-antimagic for 0 ≤ d ≤ 1 whenr = 1 and for 0 ≤ d ≤ 2r − 3 whenr ≥ 2.
Moreover, using Theorem 2.1, we can extend this result also for subdivided wheels in which not only rim edges but also spokes are subdivided.
Theorem 3.8
The subdivided wheelWn(r, s), n ≥ 5, r ≥ 1, s ≥ 0, is super (a, d)-Cr+2s+3-antimagic for 0 ≤ d ≤ 1 whenr = 1 and for 0 ≤ d ≤ 2r − 3 whenr ≥ 2.
4 Conclusion
In the present paper we showed that the property to be super (a, d)-H-antimagic is hereditary according to the operation of subdivision of edges. We proved that if a graph G is super cycle-antimagic then the subdivided graph S(G) also admits a super cycle-antimagic labeling.
This indicates that it is important to study the antimagic properties of graphs with simple structures which allows us to get result for large graphs. Recently, large graphs have attracted a lot of attention, see [15]. However, the interesting question is whether, for a given graph, it is possible to extend the set of differences also for cases not covered by the general result. It means to find a difference d such that the subdivided graph S(G) is super cycle-antimagic with the difference d but the corresponding graph G is not.
Another interesting directions for further investigation is to deal with the non-uniform subdivision and to find another graph operations that are hereditary according to being cycle-antimagic, or in general H-antimagic.
Acknowledgement
The research for this article was supported by APVV-15-0116 and by VEGA 1/0233/18.
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- On new strong versions of Browder type theorems
- A Geršgorin-type eigenvalue localization set with n parameters for stochastic matrices
- Restriction conditions on PL(7, 2) codes (3 ≤ |𝓖i| ≤ 7)
- Singular integrals with variable kernel and fractional differentiation in homogeneous Morrey-Herz-type Hardy spaces with variable exponents
- Introduction to disoriented knot theory
- Restricted triangulation on circulant graphs
- Boundedness control sets for linear systems on Lie groups
- Chen’s inequalities for submanifolds in (κ, μ)-contact space form with a semi-symmetric metric connection
- Disjointed sum of products by a novel technique of orthogonalizing ORing
- A parametric linearizing approach for quadratically inequality constrained quadratic programs
- Generalizations of Steffensen’s inequality via the extension of Montgomery identity
- Vector fields satisfying the barycenter property
- On the freeness of hypersurface arrangements consisting of hyperplanes and spheres
- Biderivations of the higher rank Witt algebra without anti-symmetric condition
- Some remarks on spectra of nuclear operators
- Recursive interpolating sequences
- Involutory biquandles and singular knots and links
- Constacyclic codes over 𝔽pm[u1, u2,⋯,uk]/〈 ui2 = ui, uiuj = ujui〉
- Topological entropy for positively weak measure expansive shadowable maps
- Oscillation and non-oscillation of half-linear differential equations with coeffcients determined by functions having mean values
- On 𝓠-regular semigroups
- One kind power mean of the hybrid Gauss sums
- A reduced space branch and bound algorithm for a class of sum of ratios problems
- Some recurrence formulas for the Hermite polynomials and their squares
- A relaxed block splitting preconditioner for complex symmetric indefinite linear systems
- On f - prime radical in ordered semigroups
- Positive solutions of semipositone singular fractional differential systems with a parameter and integral boundary conditions
- Disjoint hypercyclicity equals disjoint supercyclicity for families of Taylor-type operators
- A stochastic differential game of low carbon technology sharing in collaborative innovation system of superior enterprises and inferior enterprises under uncertain environment
- Dynamic behavior analysis of a prey-predator model with ratio-dependent Monod-Haldane functional response
- The points and diameters of quantales
- Directed colimits of some flatness properties and purity of epimorphisms in S-posets
- Super (a, d)-H-antimagic labeling of subdivided graphs
- On the power sum problem of Lucas polynomials and its divisible property
- Existence of solutions for a shear thickening fluid-particle system with non-Newtonian potential
- On generalized P-reducible Finsler manifolds
- On Banach and Kuratowski Theorem, K-Lusin sets and strong sequences
- On the boundedness of square function generated by the Bessel differential operator in weighted Lebesque Lp,α spaces
- On the different kinds of separability of the space of Borel functions
- Curves in the Lorentz-Minkowski plane: elasticae, catenaries and grim-reapers
- Functional analysis method for the M/G/1 queueing model with single working vacation
- Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions
- The existence of solutions to certain type of nonlinear difference-differential equations
- Domination in 4-regular Knödel graphs
- Stepanov-like pseudo almost periodic functions on time scales and applications to dynamic equations with delay
- Algebras of right ample semigroups
- Random attractors for stochastic retarded reaction-diffusion equations with multiplicative white noise on unbounded domains
- Nontrivial periodic solutions to delay difference equations via Morse theory
- A note on the three-way generalization of the Jordan canonical form
- On some varieties of ai-semirings satisfying xp+1 ≈ x
- Abstract-valued Orlicz spaces of range-varying type
- On the recursive properties of one kind hybrid power mean involving two-term exponential sums and Gauss sums
- Arithmetic of generalized Dedekind sums and their modularity
- Multipreconditioned GMRES for simulating stochastic automata networks
- Regularization and error estimates for an inverse heat problem under the conformable derivative
- Transitivity of the εm-relation on (m-idempotent) hyperrings
- Learning Bayesian networks based on bi-velocity discrete particle swarm optimization with mutation operator
- Simultaneous prediction in the generalized linear model
- Two asymptotic expansions for gamma function developed by Windschitl’s formula
- State maps on semihoops
- 𝓜𝓝-convergence and lim-inf𝓜-convergence in partially ordered sets
- Stability and convergence of a local discontinuous Galerkin finite element method for the general Lax equation
- New topology in residuated lattices
- Optimality and duality in set-valued optimization utilizing limit sets
- An improved Schwarz Lemma at the boundary
- Initial layer problem of the Boussinesq system for Rayleigh-Bénard convection with infinite Prandtl number limit
- Toeplitz matrices whose elements are coefficients of Bazilevič functions
- Epi-mild normality
- Nonlinear elastic beam problems with the parameter near resonance
- Orlicz difference bodies
- The Picard group of Brauer-Severi varieties
- Galoisian and qualitative approaches to linear Polyanin-Zaitsev vector fields
- Weak group inverse
- Infinite growth of solutions of second order complex differential equation
- Semi-Hurewicz-Type properties in ditopological texture spaces
- Chaos and bifurcation in the controlled chaotic system
- Translatability and translatable semigroups
- Sharp bounds for partition dimension of generalized Möbius ladders
- Uniqueness theorems for L-functions in the extended Selberg class
- An effective algorithm for globally solving quadratic programs using parametric linearization technique
- Bounds of Strong EMT Strength for certain Subdivision of Star and Bistar
- On categorical aspects of S -quantales
- On the algebraicity of coefficients of half-integral weight mock modular forms
- Dunkl analogue of Szász-mirakjan operators of blending type
- Majorization, “useful” Csiszár divergence and “useful” Zipf-Mandelbrot law
- Global stability of a distributed delayed viral model with general incidence rate
- Analyzing a generalized pest-natural enemy model with nonlinear impulsive control
- Boundary value problems of a discrete generalized beam equation via variational methods
- Common fixed point theorem of six self-mappings in Menger spaces using (CLRST) property
- Periodic and subharmonic solutions for a 2nth-order p-Laplacian difference equation containing both advances and retardations
- Spectrum of free-form Sudoku graphs
- Regularity of fuzzy convergence spaces
- The well-posedness of solution to a compressible non-Newtonian fluid with self-gravitational potential
- On further refinements for Young inequalities
- Pretty good state transfer on 1-sum of star graphs
- On a conjecture about generalized Q-recurrence
- Univariate approximating schemes and their non-tensor product generalization
- Multi-term fractional differential equations with nonlocal boundary conditions
- Homoclinic and heteroclinic solutions to a hepatitis C evolution model
- Regularity of one-sided multilinear fractional maximal functions
- Galois connections between sets of paths and closure operators in simple graphs
- KGSA: A Gravitational Search Algorithm for Multimodal Optimization based on K-Means Niching Technique and a Novel Elitism Strategy
- θ-type Calderón-Zygmund Operators and Commutators in Variable Exponents Herz space
- An integral that counts the zeros of a function
- On rough sets induced by fuzzy relations approach in semigroups
- Computational uncertainty quantification for random non-autonomous second order linear differential equations via adapted gPC: a comparative case study with random Fröbenius method and Monte Carlo simulation
- The fourth order strongly noncanonical operators
- Topical Issue on Cyber-security Mathematics
- Review of Cryptographic Schemes applied to Remote Electronic Voting systems: remaining challenges and the upcoming post-quantum paradigm
- Linearity in decimation-based generators: an improved cryptanalysis on the shrinking generator
- On dynamic network security: A random decentering algorithm on graphs
