# Total edge irregularity strength of trees

• Volume: 26, Issue: 3, page 449-456
• ISSN: 2083-5892

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## Abstract

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A total edge-irregular k-labelling ξ:V(G)∪ E(G) → {1,2,...,k} of a graph G is a labelling of vertices and edges of G in such a way that for any different edges e and f their weights wt(e) and wt(f) are distinct. The weight wt(e) of an edge e = xy is the sum of the labels of vertices x and y and the label of the edge e. The minimum k for which a graph G has a total edge-irregular k-labelling is called the total edge irregularity strength of G, tes(G). In this paper we prove that for every tree T of maximum degree Δ on p vertices tes(T) = max{⎡(p+1)/3⎤,⎡(Δ+1)/2⎤}.

## How to cite

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Jaroslav Ivančo, and Stanislav Jendrol'. "Total edge irregularity strength of trees." Discussiones Mathematicae Graph Theory 26.3 (2006): 449-456. <http://eudml.org/doc/270466>.

@article{JaroslavIvančo2006,
abstract = { A total edge-irregular k-labelling ξ:V(G)∪ E(G) → \{1,2,...,k\} of a graph G is a labelling of vertices and edges of G in such a way that for any different edges e and f their weights wt(e) and wt(f) are distinct. The weight wt(e) of an edge e = xy is the sum of the labels of vertices x and y and the label of the edge e. The minimum k for which a graph G has a total edge-irregular k-labelling is called the total edge irregularity strength of G, tes(G). In this paper we prove that for every tree T of maximum degree Δ on p vertices tes(T) = max\{⎡(p+1)/3⎤,⎡(Δ+1)/2⎤\}. },
author = {Jaroslav Ivančo, Stanislav Jendrol'},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {graph labelling; tree; irregularity strength; total labellings; total edge irregularity strength},
language = {eng},
number = {3},
pages = {449-456},
title = {Total edge irregularity strength of trees},
url = {http://eudml.org/doc/270466},
volume = {26},
year = {2006},
}

TY - JOUR
AU - Jaroslav Ivančo
AU - Stanislav Jendrol'
TI - Total edge irregularity strength of trees
JO - Discussiones Mathematicae Graph Theory
PY - 2006
VL - 26
IS - 3
SP - 449
EP - 456
AB - A total edge-irregular k-labelling ξ:V(G)∪ E(G) → {1,2,...,k} of a graph G is a labelling of vertices and edges of G in such a way that for any different edges e and f their weights wt(e) and wt(f) are distinct. The weight wt(e) of an edge e = xy is the sum of the labels of vertices x and y and the label of the edge e. The minimum k for which a graph G has a total edge-irregular k-labelling is called the total edge irregularity strength of G, tes(G). In this paper we prove that for every tree T of maximum degree Δ on p vertices tes(T) = max{⎡(p+1)/3⎤,⎡(Δ+1)/2⎤}.
LA - eng
KW - graph labelling; tree; irregularity strength; total labellings; total edge irregularity strength
UR - http://eudml.org/doc/270466
ER -

## References

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8. [8] A. Frieze, R.J. Gould, M. Karoński and F. Pfender, On graph irregularity strength, J. Graph Theory 41 (2002) 120-137, doi: 10.1002/jgt.10056. Zbl1016.05045
9. [9] J.A. Gallian, Graph labeling, The Electronic Jounal of Combinatorics, Dynamic Survey DS6 (October 19, 2003).
10. [10] J. Lehel, Facts and quests on degree irregular assignment, in: Graph Theory, Combin. Appl. vol. 2, Y. Alavi, G. Chartrand, O.R. Oellermann and A.J. Schwenk, eds., (John Wiley and Sons, Inc., 1991) 765-782. Zbl0841.05052
11. [11] T. Nierhoff, A tight bound on the irregularity strength of graphs, SIAM J. Discrete Math. 13 (2000) 313-323, doi: 10.1137/S0895480196314291. Zbl0947.05067
12. [12] W. D. Wallis, Magic Graphs (Birkhäuser Boston, 2001), doi: 10.1007/978-1-4612-0123-6. Zbl0979.05001

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