More classes of super cycle-antimagic graphs

Journal of Discrete Mathematical Sciences and Cryptography, 2012

A simple graph G = (V, E) admits an H-covering if every edge in E belongs to at least one subgraph of G isomorphic to a given graph H. The graph G admitting an H-covering is (a, d)-H-antimagic if there exists a bijection f : V ∪ E → {1, 2,. .. , |V | + |E|} such that, for all subgraphs H ′ of G isomorphic to H, the H ′-weights, wt f (H ′) = ∑ v∈V (H ′) f (v) + ∑ e∈E(H ′) f (e), form an arithmetic progression with the initial term a and the common difference d. Such a labeling is called super if the smallest possible labels appear on the vertices. In this paper we prove the existence of super (a, d)-H-antimagic labelings of fan graphs and ladders for H isomorphic to a cycle.

Hacettepe Journal of Mathematics and Statistics, 2019

A simple graph $G=(V,E)$ admits an~$H$-covering if every edge in $E$ belongs to at least one subgraph of $G$ isomorphic to a given graph $H$. The graph $G$ admitting an $H$-covering is $(a,d)$-$H$-antimagic if there exists a~bijection $f:V\cup E\to\{1,2,\cdots,|V|+|E|\}$ such that, for all subgraphs $H'$ of $G$ isomorphic to $H$, the $H'$-weights, $wt_f(H')= \sum_{v\in V(H')} f(v) + \sum_{e\in E(H')} f(e)$, form an~arithmetic progression with the initial term $a$ and the common difference $d$. Such a labeling is called {\it super} if the smallest possible labels appear on the vertices. In this paper we prove the existence of super $(a,d)$-$H$-antimagic labelings of fan graphs and ladders for $H$ isomorphic to a cycle.

Australas. J Comb., 2017

A simple graph G = (V,E) admits an H-covering if every edge in E belongs to a subgraph of G isomorphic to a given graph H . Then the graph G admitting an H-covering is (a, d)-H-antimagic if there exists a bijection f : V ∪ E → {1, 2, . . . , |V | + |E|} such that, for all subgraphs H ′ of G isomorphic to H , the H ′-weights, wtf(H ′) = ∑ v∈V (H′) f(v) + ∑ e∈E(H′) f(e), form an arithmetic progression a, a + d, a + 2d, . . . , a + (t − 1)d where a is the first term, d is the common difference and t is the number of subgraphs of G isomorphic to H . Such a labeling is called super if f(V ) = {1, 2, . . . , |V |}. This paper deals with some results on anti-balanced sets and we show the existence of super (a, d)-cycle-antimagic labelings of fans and some square graphs.

Acta Mechanica Slovaca

Let G = (V,E) be a finite simple graph with p vertices and q edges. An edge-covering of G is a family of subgraphs H1,H2, ... ,Ht such that each edge of E(G) belongs to at least one of the subgraphs Hi, i=1,2, ... ,t. If every subgraph Hi is isomorphic to a given graph H, then the graph G admits an H-covering. Such a graph G is called (a,d)-H-antimagic if there is a bijection f: VjEg{1,2, ... ,p+q} such that for all subgraphs H′ of G isomorphic to H, the sum of the labels of all the edges and vertices belonging to H′ constitutes an arithmetic progression with the initial term a and the common difference d. When f(V)={1,2, ... ,p}, then G is said to be super (a,d)-H-antimagic; and if d = 0 then G is called H-supermagic. We will exhibit an operation on graphs which keeps super H-antimagic properties. We use a technique of partitioning sets of integers for the construction of the required labelings.

Open Mathematics

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.

Hacettepe Journal of Mathematics and Statistics,

A simple graph G = (V, E) admits an H-covering if every edge in E belongs to at least one subgraph of G isomorphic to a given graph H. The graph G admitting an H-covering is (a, d)-H-antimagic if there exists a bijection f : V ∪ E → {1, 2,. .. , |V | + |E|} such that, for all subgraphs H ′ of G isomorphic to H, the H ′-weights, wt f (H ′) = ∑ v∈V (H ′) f (v) + ∑ e∈E(H ′) f (e), form an arithmetic progression with the initial term a and the common difference d. Such a labeling is called super if the smallest possible labels appear on the vertices. In this paper we prove the existence of super (a, d)-H-antimagic labelings of fan graphs and ladders for H isomorphic to a cycle.

International Journal of Mathematics and Soft Computing

A simple graph G = (V, E) admits an H-covering if every edge in E belongs to a subgraph of G isomorphic to H. G is H-magic if there is a total labeling f : V ∪ E → {1, 2, 3, • • • , |V | + |E|} such that for each subgraph H = (V , E) of G isomorphic to H, v∈V1 f (v) + e∈E1 f (e) = s is constant. When f (V) = {1, 2, • • • , |V |}, then G is said to be H-supermagic. In this paper, we show that P m,n and the splitting graph of a cycle C n are cycle-supermagic.

TURKISH JOURNAL OF MATHEMATICS, 2018

A graph G = (V (G), E(G)) admits an H-covering if every edge in E belongs to a subgraph of G isomorphic to H. A graph G admitting an H-covering is called (a, d)-H-antimagic if there is a bijection f : V (G) ∪ E(G) → {1, 2,. .. , |V (G)| + |E(G)|} such that, for all subgraphs H ′ of G isomorphic to H , the H-weights, wt f (H ′) = ∑ v∈V (H ′) f (v)+ ∑ e∈E(H ′) f (e), constitute an arithmetic progression with the initial term a and the common difference d. In this paper we provide some sufficient conditions for the Cartesian product of graphs to be H-antimagic. We use partitions subsets of integers for describing desired H-antimagic labelings.

International Journal of Engineering and Technology, 2017

A graph G of order p and size q is called (a,d)-edge-antimagic total if there exists a one-to-one and onto mapping f from ∪ , , … , such that the edge weights , ∈ form an AP progression with first term 'a' and common difference 'd'. The graph G is said to be Super (a,d)-edge-antimagic total labeling if the , , … , . In this paper we obtain Super (a,d)-edge-antimagic properties of certain classes of graphs, including Fans graph, Single fan graph, Half Kite graph and Ambrela graph.

AKCE International Journal of Graphs and Combinatorics, 2018

Let H and G be finite simple graphs where every edge of G belongs to at least one subgraph that is isomorphic to H. An (a,d)-H-antimagic total labeling of a graph G is a bijection f:V(G)∪E(G)→{1,2,…,|V(G)|+|E(G)|} such that for all subgraphs H′ isomorphic to H, the H-weights, w(H′)=∑v∈V(H′)f(v)+∑uv∈E(H′)f(uv) form an arithmetic progression {a,a+d,…,a+(k−1)d} where a>0,d≥0 are two fixed integers and k is the number of subgraphs of G isomorphic to H. Moreover, if the vertex set V(G) receives the minimum possible labels {1,2,…,|V(G)|}, then f is called a super (a,d)-H-antimagic total labeling. In this paper we study super (a,d)-Cn-antimagic total labeling of a disconnected graph, namely mCn.