Irregular labelings of circulant graphs

For a simple graph G=(V(G),E(G)), a total labeling f: V(G)jE(G) g{1,2, ... ,k} is called k-labeling. The weight of an edge xy in G, denoted by wtf(xy), is the sum of the edge label itself and the labels of end vertices x and y, i.e. wtf(xy)=f(x)+f(xy)+f(y). A total k-labeling is defined to be an edge irregular total k-labeling of the graph G if for every two different edges xy and x′y′ there is wtf(xy)≠wtf(x′y′). The minimum k for which the graph G has an edge irregular total k-labeling is called the total edge irregularity strength of G, denoted by tes(G). In this paper, we estimate the upper bound of the total edge irregularity strength of disjoint union of multiple copies of a graph and we prove that this upper bound is tight.

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“…Finding the irregularity strength of a graph seems to be hard even for graphs with simple structure, see [5,7,13,16,17,22,24,26].…”

Section: Definitions and Known Resultsmentioning

confidence: 99%

On the Total Edge Irregularity Strength of Disjoint Union of Graphs

Bača¹,

Ryan²,

Semaničová-Feňovčíková³

2015

AMS

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“…These results were then improved by Przyby lo [22], Anholcer et al [3] and Nurdin et al [21]. Other interesting results on total vertex (edge) irregularity strengths can be found in [1,2,4,11,14,15,17,19].…”

Section: Introductionmentioning

confidence: 88%

On face irregular evaluations of plane graphs

Bača¹,

Ovais²,

Semaničová-Feňovčíková³

et al. 2020

Discuss. Math. Graph Theory

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“…The concept on nowhere-zero modular edge-gracefulness was already considered in [17]. The following open problem was stated in [6]. For any Abelian group G, let G * = G \ {0}.…”

Section: Theorem 12 ([8])mentioning

confidence: 99%

“…Irregularity strength of disconnected graphs was considered in [6,8]. In particular, the following exponential upper bound on s g (G) was provided there.…”

Section: Introductionmentioning

confidence: 99%

Linear bounds on nowhere-zero group irregularity strength and nowhere-zero group sum chromatic number of graphs

Anholcer¹,

Cichacz²,

Przybyło³

2019

Applied Mathematics and Computation

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We investigate the group irregularity strength, s g (G), of a graph, i.e. the least integer k such that taking any Abelian group G of order k, there exists a function f : E(G) → G so that the sums of edge labels incident with every vertex are distinct. So far the best upper bound on s g (G) for a general graph G was exponential in n − c, where n is the order of G and c denotes the number of its components. In this note we prove that s g (G) is linear in n, namely not greater than 2n. In fact, we prove a stronger result, as we additionally forbid the identity element of a group to be an edge label or the sum of labels around a vertex. We consider also locally irregular labelings where we require only sums of adjacent vertices to be distinct. For the corresponding graph invariant we prove the general upper bound: ∆(G) + col(G) − 1 (where col(G) is the coloring number of G) in the case when we do not use the identity element as an edge label, and a slightly worse one if we additionally forbid it as the sum of labels around a vertex. In the both cases we also provide a sharp upper bound for trees and a constant upper bound for the family of planar graphs.

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Irregular labelings of circulant graphs

Cited by 38 publications

References 10 publications

On the Total Edge Irregularity Strength of Disjoint Union of Graphs

On the Total Edge Irregularity Strength of Disjoint Union of Graphs

On face irregular evaluations of plane graphs

Linear bounds on nowhere-zero group irregularity strength and nowhere-zero group sum chromatic number of graphs

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