Discussiones Mathematicae Graph Theory 26(3) (2006) 449-456
doi: 10.7151/dmgt.1337

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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
tes(T) = max
{⎡(p+1)/3⎤, ⎡(Δ+1)/2⎤}.

Keywords: graph labelling, tree, irregularity strength, total labellings, total edge irregularity strength.

2000 Mathmatics Subject Classification: 05C78, 05C05.

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Received 30 September 2005
Revised 31 January 2006