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On the total vertex irregularity strength of trees. (English) Zbl 1208.05014
Summary: A vertex irregular total \(k\)-labelling \(\lambda :V(G) \cup E(G) \to \{1,2,\dots ,k\}\) of a graph \(G\) is a labelling of vertices and edges of \(G\) done in such a way that for any different vertices \(x\) and \(y\), their weights \(wt(x)\) and \(wt(y)\) are distinct. The weight \(wt(x)\) of a vertex \(x\) is the sum of the label of \(x\) and the labels of all edges incident with \(x\). The minimum \(k\) for which a graph \(G\) has a vertex irregular total \(k\)-labelling is called the total vertex irregularity strength of \(G\), denoted by tvs\((G)\). In this paper, we determine the total vertex irregularity strength of trees.

MSC:
05C05 Trees
05C78 Graph labelling (graceful graphs, bandwidth, etc.)
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[1] Amar, D.; Togni, O., Irregularity strength of trees, Discrete math., 190, 15-38, (1998) · Zbl 0956.05092
[2] Anholcer, M.; Kalkowski, M.; Przybyło, J., A new upper bound for the total vertex irregularity strength of graphs, Discrete math., 309, 6316-6317, (2009) · Zbl 1210.05117
[3] M. Bača, S. Jendrol’, M. Miller, On total edge irregular labellings of trees (unpublished).
[4] Bača, M.; Jendrol’, S.; Miller, M.; Ryan, J., On irregular total labellings, Discrete math., 307, 1378-1388, (2007) · Zbl 1115.05079
[5] Bohman, T.; Kravitz, D., On the irregularity strength of trees, J. graph theory, 45, 4, 241-254, (2004) · Zbl 1034.05015
[6] Chartrand, G.; Jacobson, M.S.; Lehel, J.; Oellermann, O.R.; Ruiz, S.; Saba, F., Irregular networks, Congr. numer., 64, 187-192, (1988)
[7] Gallian, J.A., Graph labeling, Electron. J. combin. dyn. surv., DS6, (2007)
[8] Ivančo, J.; Jendrol’, S., Total edge irregularity strength of trees, Discuss. math. graph theory, 26, 449-456, (2006) · Zbl 1135.05066
[9] M.S. Jacobson, J. Lehel, Degree irregularity. Available online at: http://athena.luisville.edu/ msjaco01/irregbib.html.
[10] Nierhoff, T., A tight bound on the irregularity strength of graphs, SIAM J. discrete math., 13, 3, 313-323, (2000) · Zbl 0947.05067
[11] Nurdin; Baskoro, E.T.; Salman, A.N.M., The total edge irregular strengths of union graphs of \(K_{2, n}\), J. mat. sain, 11, 3, 105-109, (2006)
[12] Nurdin, E.T. Baskoro, A.N.M. Salman, N.N. Gaos, On total vertex-irregular labellings for several types of trees, Util. Math. (in press). · Zbl 1242.05242
[13] Nurdin; Salman, A.N.M.; Baskoro, E.T., The total edge-irregular strengths of the corona product of paths with some graphs, J. combin. math. combin. comput., 65, 163-175, (2008) · Zbl 1172.05033
[14] Wallis, W.D., Magic graphs, (2001), Birkhäuser Boston · Zbl 0979.05001
[15] Wijaya, K.; Slamin; Surahmat; Jendrol’, S., Total vertex irregular labeling of complete bipartite graphs, J. combin. math. combin. comput., 55, 129-136, (2005) · Zbl 1100.05090
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