A Note on Edge Irregularity Strength of Some Graphs

Indonesian Journal of Combinatorics, 2021

An edge irregular total k-labeling f : V ∪ E → {1, 2, ..., k} of a graph G = (V, E) is a labeling of vertices and edges of G in such a way that for any two different edges uv and u v , their weights f (u) + f (uv) + f (v) and f (u) + f (u v) + f (v) are distinct.The total edge irregularity strength tes(G) is defined as the minimum k for which the graph G has an edge irregular total k-labeling. In this paper, we determine the total edge irregularity strength of new classes of graphs C m @C n , P * m,n and C * m,n and hence we extend the validity of the conjecture tes(G) = max |E(G)|+2 3 , ∆(G)+1 2 for some more graphs.

Bulletin of the InternationalMathematical Virtual Institute, 2019

A totally irregular total k-labeling f : V ∪ E → {1, 2, 3,. .. , k} is a labeling of vertices and edges of G in such a way that for any two different vertices x and y their vertex-weights wt h (x) = wt h (y) where the vertex-weight wt h (x) = h(x) + xy∈E h(xz) and also for every two different edges xy and x ′ y ′ of G their edge-weights wt h (xy) = h(x) + h(xy) + h(y) and wt h (x ′ y ′) = h(x ′) + h(x ′ y ′) + h(y ′) are distinct. A total irregularity strength of graph G, denoted by ts(G) is defined as the minimum k for which a graph G has a totally irregular total k-labeling. In this paper, we investigate double fan, double triangular snake, joint-wheel and Pm + Km whose total irregularity strength equals to the lower bound. 2010 Mathematics Subject Classification. 05C78. Key words and phrases. vertex irregular total k-labeling; edge irregular total k-labeling; total irregularity strength;double fan graph;double triangular snake graph; joint-wheel graph.

Discussiones Mathematicae Graph Theory, 2019

For a graph G an edge-covering of G is a family of subgraphs H 1 , H 2 ,. .. , H t such that each edge of E(G) belongs to at least one of the subgraphs H i , i = 1, 2,. .. , t. In this case we say that G admits an (H 1 , H 2 ,. .. , H t)-(edge) covering. An H-covering of graph G is an (H 1 , H 2 ,. .. , H t)-(edge) covering in which every subgraph H i is isomorphic to a given graph H. Let G be a graph admitting H-covering. An edge k-labeling α : E(G) → {1, 2,. .. , k} is called an H-irregular edge k-labeling of the graph G if for every two different subgraphs H ′ and H ′′ isomorphic to H their weights 1 Corresponding author. 2 Naeem, Siddiqui, Bača, Semaničová-Feňovčíková and Ashraf wt α (H ′) and wt α (H ′′) are distinct. The weight of a subgraph H under an edge k-labeling α is the sum of labels of edges belonging to H. The edge H-irregularity strength of a graph G, denoted by ehs(G, H), is the smallest integer k such that G has an H-irregular edge k-labeling. In this paper we determine the exact values of ehs(G, H) for prisms, antiprisms, triangular ladders, diagonal ladders, wheels and gear graphs. Moreover the subgraph H is isomorphic to only C 4 , C 3 and K 4 .

Applied Mathematical Sciences, 2019

Let G = (V, E) be a graph and f be a vertex k-labeling of G. The weight of an edge uv ∈ E with respect to f , denoted by w f (uv), is given by w f (uv) = f (u) + f (v). A vertex k-labeling f is said to be an edge r-irregular k-labeling if at most r edges are permitted to have the same weight. The minimum k for which the graph G has an edge r-irregular k-labeling is called the edge r-irregularity strength of G, denoted by es r (G). In this paper, we gave the edge r-irregularity strength of paths, present a tight upper-bound for the edge r-irregularity strength of cycles, and state some characterizations regarding the edge r-irregularity strength of complete graphs.

Indonesian Journal of Combinatorics, 2020

Under a totally irregular total k-labeling of a graph G = (V, E), we found that for some certain graphs, the edge-weight set W (E) and the vertex-weight set W (V) of G which are induced by k = ts(G), W (E) ∩ W (V) is a nonempty set. For which k, a graph G has a totally irregular total labeling if W (E) ∩ W (V) = ∅? We introduce the total disjoint irregularity strength, denoted by ds(G), as the minimum value k where this condition satisfied. We provide the lower bound of ds(G) and determine the total disjoint irregularity strength of cycles, paths, stars, and complete graphs.

International Journal of Computer Applications, 2012

  where N(x) is the set of neighbours of x. f is called a vertex irregular total k-labeling if for every pair of distinct vertices x and y () f wt x  () f wt y. The minimum k for which the graph G has a vertex irregular total k-labeling is called the total vertex irregularity strength of G and is denoted by tvs(G). In this paper we find the total vertex irregularity strength of cycle related graphs H(n), DHF(n), F(n,2) and obtain a bound for the total vertex irregularity strength of H graphs H(k).

2018

Abstract. A vertex irregular total k-labeling of a graph G with vertex setV and edge set E is an assignment of positive integer labels {1,2, ..., k} to both vertices and edges so that the weights calculated at vertices are distinct. The total vertex irregularity strength of G, denoted bytvs(G) is the minimum value of the largest label k over all such irregular assignment. In this paper, we study the total vertex irregularity strength of cycle quadrilateral snake, s unflower, double wheel, fungus, triangular book and quadrilateral book.

Journal of The International Mathematical Virtual Insitute, 2019

An edge irregular total k-labeling f : V ∪ E → {1, 2, 3, . . . , k} ofa graph G = (V, E) is a labeling of vertices and edges of G in such a way thatfor any two different edges uv and u′v′ their weights f (u) + f (uv) + f (v) andf (u′) + f (u′v′) + f (v′) are distinct. The total edge irregularity strength tes(G)is defined as the minimum k for which the graph G has an edge irregular totalk-labeling. In this paper, we study the total edge irregularity strength forshadow graph of cycle and path, total graph of cycle and path, lotus inside acircle, double wheel graph (

Mathematics in Computer Science, 2015

A total k-labeling is a map that carries vertices and edges of a graph G into a set of positive integer labels {1, 2, ..., k}. An edge irregular total k-labeling of a graph G is a total k-labeling such that the weights calculated for all edges are distinct. The weight of an edge uv in G is defined as the sum of the label of u, the label of v and the label of uv. The total edge irregularity strength of G, denoted by tes(G), is the minimum value of the largest label k over all such edge irregular total k-labelings. In this paper, we investigate the total edge irregularity strength of generalized helm, H m n for n ≥ 3, m = 1, 2, and m ≡ 0 (mod 3).

AIP Conference Proceedings

Let G be an undirected, simple, nontrivial, and finite graphs admitting an-covering. The total-labeling is called a total-irregular-labeling of if for any pair of subgraphs and , it holds when. Define an-weight, denoted by , which sum of all edge and vertex labels in subgraph under the total-labeling. The smallest positive integer such that has an-irregular totallabeling is the total-irregularity strength of , denoted by. A-tadpole graph is a graph on vertices and denoted by. In this paper, we find this of some graphs with-covering, i.e. generalized butterfly graphs, and eclipse graphs.