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⎤}.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1337,
author = {Jaroslav Ivan\v co and Stanislav Jendrol'},
title = {Total edge irregularity strength of trees},
journal = {Discussiones Mathematicae Graph Theory},
volume = {26},
year = {2006},
pages = {449-456},
zbl = {1135.05066},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1337}
}
Jaroslav Ivančo; Stanislav Jendrol'. Total edge irregularity strength of trees. Discussiones Mathematicae Graph Theory, Tome 26 (2006) pp. 449-456. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1337/
[000] [1] M. Aigner and E. Triesch, Irregular assignment of trees and forests, SIAM J. Discrete Math. 3 (1990) 439-449, doi: 10.1137/0403038. | Zbl 0735.05049
[001] [2] D. Amar and O. Togni, Irregularity strength of trees, Discrete Math. 190 (1998) 15-38, doi: 10.1016/S0012-365X(98)00112-5. | Zbl 0956.05092
[002] [3] M. Bača, S. Jendrol' and M. Miller, On total edge irregular labelling of trees, (submitted).
[003] [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. | Zbl 1115.05079
[004] [5] T. Bohman and D. Kravitz, On the irregularity strength of trees, J. Graph Theory 45 (2004) 241-254, doi: 10.1002/jgt.10158. | Zbl 1034.05015
[005] [6] L.A. Cammack, R.H. Schelp and G.C. Schrag, Irregularity strength of full d-ary trees, Congr. Numer. 81 (1991) 113-119. | Zbl 0765.05037
[006] [7] G. Chartrand, M.S. Jacobson, J. Lehel, O.R. Oellermann, S. Ruiz and F. Saba, Irregular networks, Congr. Numer. 64 (1988) 187-192.
[007] [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. | Zbl 1016.05045
[008] [9] J.A. Gallian, Graph labeling, The Electronic Jounal of Combinatorics, Dynamic Survey DS6 (October 19, 2003).
[009] [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. | Zbl 0841.05052
[010] [11] T. Nierhoff, A tight bound on the irregularity strength of graphs, SIAM J. Discrete Math. 13 (2000) 313-323, doi: 10.1137/S0895480196314291. | Zbl 0947.05067
[011] [12] W. D. Wallis, Magic Graphs (Birkhäuser Boston, 2001), doi: 10.1007/978-1-4612-0123-6. | Zbl 0979.05001