Total Face Irregularity Strength of Grid and Wheel Graph under K-Labeling of Type (1, 1, 0)

Aleem Mughal,Noshad Jamil,Muhammad Kamran Siddiqui

doi:10.1155/2021/1311269

Aleem Mughal, Noshad Jamil + Show 1 more

Open Access

https://doi.org/10.1155/2021/1311269

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Abstract

In this study, we used grids and wheel graphs G = V , E , F , which are simple, finite, plane, and undirected graphs with V as the vertex set, E as the edge set, and F as the face set. The article addresses the problem to find the face irregularity strength of some families of generalized plane graphs under k -labeling of type α , β , γ . In this labeling, a graph is assigning positive integers to graph vertices, graph edges, or graph faces. A minimum integer k for which a total label of all verteices and edges of a plane graph has distinct face weights is called k -labeling of a graph. The integer k is named as total face irregularity strength of the graph and denoted as tfs G . We also discussed a special case of total face irregularity strength of plane graphs under k -labeling of type (1, 1, 0). The results will be verified by using figures and examples.

Highlights

A k-labeling φ of type (α, β, c) of the plane graph G is called face irregular k-labeling of type (α, β, c) of the plane graph G if every two different faces have distinct weights; that is, for graph faces f, g ∈ G and f ≠ g, we have
For a vertex-edge labeled graph G, the minimum integer k for which the graph G admits a face irregular k-labeling of type (α, β, c) is called the total face irregularity strength of type (α, β, c) of the plane graph G, and it is denoted by tfs(α,β,c)(G)
Baca et al determined a lower bound for the face irregularity strength of type (α, β, c) when a 2-connected plane graph G has more than one faces of the largest sizes [14, 16]. ey presented the following theorem

Summary

Aleem Mughal and Noshad Jamil

We used grids and wheel graphs G (V, E, F), which are simple, finite, plane, and undirected graphs with V as the vertex set, E as the edge set, and F as the face set. e article addresses the problem to find the face irregularity strength of some families of generalized plane graphs under k-labeling of type (α, β, c). For a vertex-edge labeled graph G, the minimum integer k for which the graph G admits a face irregular k-labeling of type (α, β, c) is called the total face irregularity strength of type (α, β, c) of the plane graph G, and it is denoted by tfs(α,β,c)(G). We will calculate the total face irregularity strength of grid graphs under labeling φ of type (α, β, c), and this work is a modification of abovementioned articles. We will prove the exact value for the total face irregularity strength under k-labeling φ of type (α, β, c) of wheel graph Wn. Baca et al determined a lower bound for the face irregularity strength of type (α, β, c) when a 2-connected plane graph G has more than one faces of the largest sizes [14, 16]. We present the following theorem to calculate the lower bounds for grid graphs Gmn

Main Results

For i

Wn Wn

Weight of interior faces will be

Conclusion