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The irregularity strength of \(K_{m,m}\) is 4 for odd m. (English) Zbl 0681.05066
G. Chartrand et al. showed that for odd m, \(m\geq 3\), the edges of \(K_{m,m}\) can be labelled with 1, 2, 3, 4 in such a way that the (weighted) degrees of the vertices are all different. They conjectured that no such labelling exists with labels 1, 2, 3. In this note we prove this conjecture. G. Chartrand et al. showed that for odd m, \(m\geq 3\), the edges of \(K_{m,m}\) can be labelled with 1, 2, 3, 4 in such a way that the (weighted) degrees of the vertices are all different. They conjectured that no such labelling exists with labels 1, 2, 3. In this note we prove this conjecture.
Reviewer: R.C.Read

MSC:
05C99 Graph theory
Keywords:
edge-labelling
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References:
[1] G. Chartrand, M. Jacobson, J. Lebel, O. Oellerman, S. Ruiz and F. Saba, Irregular networks, Submitted to Fort Wayne Conference Proceedings.
[2] R. Faudree, M. Jacobson, J. Lehel and R. H. Schelp, Irregular networks, Regular graphs and integer matrices with distinct row and column sums, Submitted to Discrete Math. · Zbl 0685.05029
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