Discussiones Mathematicae Graph Theory 26(3) (2006)
449-456
DOI: https://doi.org/10.7151/dmgt.1337
TOTAL EDGE IRREGULARITY STRENGTH OF TREES
Jaroslav Ivanco and Stanislav Jendrol'
Institute of Mathematics
P.J. Safarik University
Jesenna 5, SK-041 54 Košice, Slovak Republic
e-mail: ivanco@science.upjs.sk
e-mail: jendrol@Košice.upjs.sk
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
|
Keywords: graph labelling, tree, irregularity strength, total labellings, total edge irregularity strength.
2000 Mathmatics Subject Classification: 05C78, 05C05.
References
[1] | M. Aigner and E. Triesch, Irregular assignment of trees and forests, SIAM J. Discrete Math. 3 (1990) 439-449, doi: 10.1137/0403038. |
[2] | D. Amar and O. Togni, Irregularity strength of trees, Discrete Math. 190 (1998) 15-38, doi: 10.1016/S0012-365X(98)00112-5. |
[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. |
[5] | T. Bohman and D. Kravitz, On the irregularity strength of trees, J. Graph Theory 45 (2004) 241-254, doi: 10.1002/jgt.10158. |
[6] | L.A. Cammack, R.H. Schelp and G.C. Schrag, Irregularity strength of full d-ary trees, Congr. Numer. 81 (1991) 113-119. |
[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. |
[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. |
[11] | T. Nierhoff, A tight bound on the irregularity strength of graphs, SIAM J. Discrete Math. 13 (2000) 313-323, doi: 10.1137/S0895480196314291. |
[12] | W. D. Wallis, Magic Graphs (Birkhäuser Boston, 2001), doi: 10.1007/978-1-4612-0123-6. |
Received 30 September 2005
Revised 31 January 2006
Close