Local Inclusive Distance Vertex Irregular Graphs

Martin Bača,Denny Riama Silaban,Andrea Semaničová-Feňovčíková,Kiki Ariyanti Sugeng

doi:10.3390/math9141673

Martin Bača, Denny Riama Silaban + Show 2 more

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https://doi.org/10.3390/math9141673

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Abstract

Let G=(V,E) be a simple graph. A vertex labeling f:V(G)→{1,2,⋯,k} is defined to be a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of a graph G if for any two adjacent vertices x,y∈V(G) their weights are distinct, where the weight of a vertex x∈V(G) is the sum of all labels of vertices whose distance from x is at most d (respectively, at most d but at least 1). The minimum k for which there exists a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of G is called the local inclusive (respectively, non-inclusive) d-distance vertex irregularity strength of G. In this paper, we present several basic results on the local inclusive d-distance vertex irregularity strength for d=1 and determine the precise values of the corresponding graph invariant for certain families of graphs.

Highlights

All graphs considered in this paper are simple finite
Motivated by a distance vertex labeling [8], an irregular labeling [2] and a recent paper on a local antimagic labeling [14], we introduce in this paper the concept of local inclusive and local non-inclusive d-distance vertex irregular labelings
We present several basic results and some estimations on the local inclusive 1-distance vertex irregularity strength and determine the precise values of the corresponding graph invariant for several families of graphs

Summary

Introduction

All graphs considered in this paper are simple finite. We use V(G) for the vertex set and E(G) for the edge set of a graph G. In [10] is determined the inclusive 1-distance vertex irregularity strength for paths Pn, n ≡ 0 (mod 3), stars, double stars S(m, n) with m ≤ n, a lower bound for caterpillars, cycles, and wheels. K} is defined to be a local inclusive (respectively, non-inclusive) d-distance vertex irregular labeling of a graph G if for any two adjacent vertices x, y ∈ V(G) their weights are distinct, where the weight of a vertex x ∈ V(G) is the sum of all labels of vertices whose distance from x is at most d (respectively, at most d but at least 1). We present several basic results and some estimations on the local inclusive 1-distance vertex irregularity strength and determine the precise values of the corresponding graph invariant for several families of graphs

Basic Properties

Conclusions