A new upper bound for the total vertex irregularity strength of graphs

Anholcer, Marcin; Kalkowski, Maciej; Przybyło, Jakub

doi:10.1016/j.disc.2009.05.023

Cited by 59 publications

(39 citation statements)

References 14 publications

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“…Let G be a graph of order n . We shall adjust a deterministic algorithm from in order to prove Theorem . This shall be possible if we previously order the vertices of G into a sequence

v_{1}, v_{2}, ..., v_{n}

so that for every vertex, the proportion of its neighbors preceding it in the sequence, to the total number of its neighbors is (more or less) equal to the proportion of the position of this vertex in the sequence to the total number of vertices n .…”

Section: Proofmentioning

confidence: 99%

“…Let G be a graph of order n. We shall adjust a deterministic algorithm from [2] in order to prove Theorem 1.2. This shall be possible if we previously order the vertices of G into a sequence v 1 , v 2 , .…”

Section: A General Ideamentioning

confidence: 99%

“…In it has been proven that

tvs (G) < 32 \frac{n}{δ} + 8

in general and

tvs (G) < 8 \frac{n}{r} + 3

for r ‐regular graphs. This was then improved in . Theorem Let G be a graph of order n and with minimum degree

δ > 0

.…”

Section: Introductionmentioning

confidence: 99%

“…In [14] it has been proven that tvs(G) < 32 n δ + 8 in general and tvs(G) < 8 n r + 3 for r-regular graphs. This was then improved in [2]. Theorem 1.1 ([2]).…”

Section: Introductionmentioning

confidence: 99%

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Total Vertex Irregularity Strength of Dense Graphs

Majerski

Przybyło

2013

Journal of Graph Theory

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Consider a graph G=(V,E) of minimum degree δ and order n. Its total vertex irregularity strength is the smallest integer k for which one can find a weighting w:E∪V→{1,2,...,k} such that ∑e∋uw(e)+w(u)≠∑e∋vw(e)+w(v) for every pair u,v of vertices of G. We prove that the total vertex irregularity strength of graphs with δ≥n0.5lnn is bounded from above by (2+o(1))nδ+4. One of the cornerstones of the proof is a random ordering of the vertices generated by order statistics.

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“…Let G be a graph of order n . We shall adjust a deterministic algorithm from in order to prove Theorem . This shall be possible if we previously order the vertices of G into a sequence

v_{1}, v_{2}, ..., v_{n}

Section: Proofmentioning

confidence: 99%

Section: A General Ideamentioning

confidence: 99%

“…In it has been proven that

tvs (G) < 32 \frac{n}{δ} + 8

in general and

tvs (G) < 8 \frac{n}{r} + 3

for r ‐regular graphs. This was then improved in . Theorem Let G be a graph of order n and with minimum degree

δ > 0

.…”

Section: Introductionmentioning

confidence: 99%

“…In [14] it has been proven that tvs(G) < 32 n δ + 8 in general and tvs(G) < 8 n r + 3 for r-regular graphs. This was then improved in [2]. Theorem 1.1 ([2]).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Total Vertex Irregularity Strength of Dense Graphs

Majerski

Przybyło

2013

Journal of Graph Theory

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“…In [4], Anholcer, Kalkowski and Przybylo gave the best known result for general graphs. Some other results about vertex irregular total -labeling were given by Nurdin, et al in [5] and [6], and Wijaya, et al in [7] and [8].…”

Section: ∈ ( )mentioning

confidence: 99%

On The Total Irregularity Strength of Regular Graphs

2015

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Abstract. Let = ( , ) be a graph. A total labeling : ∪ → {1, 2, ⋯ , } is called a totally irregular total -labeling of if every two distinct vertices and in satisfy ( ) ≠ ( ) and every two distinct edges 1 2 and 1 2 in satisfyThe minimum for which a graph has a totally irregular total -labeling is called the total irregularity strength of , denoted by ( ). In this paper, we consider an upper bound on the total irregularity strength of copies of a regular graph. Besides that, we give a dual labeling of a totally irregular total -labeling of a regular graph and we consider the total irregularity strength of copies of a path on two vertices, copies of a cycle, and copies of a prism 2 .

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