# Total edge irregularity strength of trees

Jaroslav Ivančo; Stanislav Jendrol'

Discussiones Mathematicae Graph Theory (2006)

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

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topJaroslav 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

top- [1] M. Aigner and E. Triesch, Irregular assignment of trees and forests, SIAM J. Discrete Math. 3 (1990) 439-449, doi: 10.1137/0403038. Zbl0735.05049
- [2] D. Amar and O. Togni, Irregularity strength of trees, Discrete Math. 190 (1998) 15-38, doi: 10.1016/S0012-365X(98)00112-5. Zbl0956.05092
- [3] M. Bača, S. Jendrol' and M. Miller, On total edge irregular labelling of trees, (submitted).
- [4] M. Bača, S. Jendrol', M. Miller and J. Ryan, On irregular total labellings, Discrete Math. 307 (2007) 1378–1388, doi: 10.1016/j.disc.2005.11.075. Zbl1115.05079
- [5] T. Bohman and D. Kravitz, On the irregularity strength of trees, J. Graph Theory 45 (2004) 241-254, doi: 10.1002/jgt.10158. Zbl1034.05015
- [6] L.A. Cammack, R.H. Schelp and G.C. Schrag, Irregularity strength of full d-ary trees, Congr. Numer. 81 (1991) 113-119. Zbl0765.05037
- [7] G. Chartrand, M.S. Jacobson, J. Lehel, O.R. Oellermann, S. Ruiz and F. Saba, Irregular networks, Congr. Numer. 64 (1988) 187-192.
- [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] J.A. Gallian, Graph labeling, The Electronic Jounal of Combinatorics, Dynamic Survey DS6 (October 19, 2003).
- [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] 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] W. D. Wallis, Magic Graphs (Birkhäuser Boston, 2001), doi: 10.1007/978-1-4612-0123-6. Zbl0979.05001

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