Super (a,d)-H-antimagic labeling of subdivided graphs

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 ∗ Corresponding author. P. JAYANTHI ET AL. /AUSTRALAS. J. COMBIN. 67 (1) (2017), 46–64 47 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.

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.

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.

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.

arXiv (Cornell University), 2015

A graph G(V, E) has an H-covering if every edge in E belongs to a subgraph of G isomorphic to H. Suppose G admits an H-covering. An H-magic labeling is a total labeling λ from V (G) ∪ E(G) onto the integers {1, 2, • • • , |V (G) ∪ E(G)|} with the property that, for every subgraph A of G isomorphic to H there is a positive integer c such that A = v∈V (A) λ(v) + e∈E(A) λ(e) = c. A graph that admits such a labeling is called H-magic. In addition, if {λ(v)}vǫV = {1, 2, • • • , |V |}, then the graph is called Hsupermagic. In this paper we formulate cycle-supermagic labelings for the disjoint union of isomorphic copies of different families of graphs. We also prove that disjoint union of non isomorphic copies of fans and ladders are cycle-supermagic.

2010

This paper deals with two types of graph labelings namely, super (a, d)-edge antimagic total labeling and (a, d)-vertex antimagic total labeling. We provide super (a, d)-edge antimagic total labeling for disjoint union Harary graphs and disjoint union of cycles. We also provide (a, d)-vertex antimagic total labeling for disjoint union of Harary graphs, disjoint union of non isomorphic cycles, sun graphs and disjoint union of non isomorphic sun graphs.

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 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.

Journal of Combinatorial Mathematics and Combinatorial Computing

Let G=(V,E) be a graph with order |G| and size |E|. An (a,d)-vertex-antimagic total labeling is a bijection α from a set of all vertices and edges to the set of consecutive integers {1,2,⋯,|V|+|E|}, such that the weights of the vertices form an arithmetic progression with the initial term a and the common difference d. If α(V(G))={1,2,⋯,|V|} then we call the labeling super (a,d)-vertex antimagic total. In this paper we show some basic properties of such labelings on a disjoint union of regular graphs and show how to construct such labelings for some classes of graphs, such as cycles, generalised Pertersen graphs and circulant graphs, for d=1.

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.