On $H$-antimagicness of Cartesian product of 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.

Journal of Physics: Conference Series, 2019

A Super (a, d) − P2 H− antimagic total labeling of a graph G = Cn H with p = |V (G)| vertices and q = |E(G)| edges is a bijective function λ from the set {V (G) ∪ E(G)} onto the set {1, 2, 3,. .. |V (G)| + |E(G)|}, such that the total P2 H−weights, wP 2 H = v∈V (P 2 H) λ(v) + e∈E(P 2 H) λ(e), form an arithmetic sequence with the smallest label appears on the vertex. This paper discusses about super (a, d) − P2 H− antimagic total labeling of disjoint union of graph G = Cn H.

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.

Bulletin of the Australian Mathematical Society, 2016

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 $H$. Then the graph $G$ is $(a,d)$-$H$-antimagic if there exists a bijection $f:V\cup E\rightarrow \{1,2,\ldots ,|V|+|E|\}$ such that, for all subgraphs $H^{\prime }$ of $G$ isomorphic to $H$, the $H^{\prime }$-weights, $wt_{f}(H^{\prime })=\sum _{v\in V(H^{\prime })}f(v)+\sum _{e\in E(H^{\prime })}f(e)$, form an arithmetic progression with the initial term $a$ and the common difference $d$. When $f(V)=\{1,2,\ldots ,|V|\}$, then $G$ is said to be super $(a,d)$-$H$-antimagic. In this paper, we study super $(a,d)$-$H$-antimagic labellings of a disjoint union of graphs for $d=|E(H)|-|V(H)|$.

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.

Heliyon, 2021

covering-shadow of a graph Closed-shadow of a graph An (,)-antimagic total labeling of a simple graph admitting an-covering is a bijection ∶ () ∪ () → {1, 2, … , | ()| + | ()|} such that for all subgraphs ′ of isomorphic to , the set of ′-weights given by (′) = ∑ ∈ (′) () + ∑ ∈ (′) () forms an arithmetic sequence , + , … , + (− 1) where > 0, ⩾ 0 are two fixed integers and is the number of all subgraphs of isomorphic to. Moreover, such a labeling is called super if the smallest possible labels appear on the vertices. A (super) (,)-antimagic graph is a graph that admits a (super) (,)-antimagic total labeling. In this paper the existence of super (,)-antimagic total labelings for the-shadow and the closed-shadow of a connected for several values of is proved.

Utilitas Mathematica

Let G=(V,E) be a graph with v vertices and e edges. An (a,d)-vertex-antimagic total labeling is a bijection λ from V(G)∪E(G) 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 common difference d. If λ(V(G))={1,2,⋯,v} then we call the labeling a super (a,d)-vertex- antimagic total. We construct (a,d)-vertex-antimagic total labelings on Harary graphs as well as or the disjoint union of k identical copies of Harary graphs.

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.

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.