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A new upper bound for the total vertex irregularity strength of graphs. (English) Zbl 1210.05117
Summary: We investigate the following modification of the well-known irregularity strength of graphs. Given a total weighting $$w$$ of a graph $$G=(V,E)$$ with elements of a set $$\{1,2,\dots ,s\}$$, denote $$wt_G(v)=\sum _{e\ni v}w(e)+w(v)$$ for each $$v\in V$$. The smallest $$s$$ for which exists such a weighting with $$wt_G(u)\neq wt_G(v)$$ whenever $$u$$ and $$v$$ are distinct vertices of $$G$$ is called the total vertex irregularity strength of this graph, and is denoted by tvs$$(G)$$. We prove that tvs$$(G)\leq 3\lceil n/\delta \rceil$$ for each graph of order $$n$$ and with minimum degree $$\delta >0$$.

##### MSC:
 05C75 Structural characterization of families of graphs 05C78 Graph labelling (graceful graphs, bandwidth, etc.)
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##### References:
 [1] Bača, M.; Jendroľ, S.; Miller, M.; Ryan, J., On irregular total labelings, Discrete math., 307, 1378-1388, (2007) · Zbl 1115.05079 [2] Chartrand, G.; Jacobson, M.S.; Lehel, J.; Oellermann, O.R.; Ruiz, S.; Saba, F., Irregular networks, Congr. numer., 64, 187-192, (1988) [3] Faudree, R.J.; Lehel, J., Bound on the irregularity strength of regular graphs, (), 247-256 · Zbl 0697.05048 [4] Frieze, A.; Gould, R.J.; Karoński, M.; Pfender, F., On graph irregularity strength, J. graph theory, 41, 2, 120-137, (2002) · Zbl 1016.05045 [5] M. Kalkowski, A note on 1, 2-conjecture, Electron. J. Combin. (submitted for publication) [6] Nierhoff, T., A tight bound on the irregularity strength of graphs, SIAM J. discrete math., 13, 3, 313-323, (2000) · Zbl 0947.05067 [7] Przybyło, J., Irregularity strength of regular graphs, Electron. J. combin., 15, 1, #R82, (2008) · Zbl 1163.05329 [8] Przybyło, J., Linear bound on the irregularity strength and the total vertex irregularity strength of graphs, SIAM J. discrete math., 23, 1, 511-516, (2009) · Zbl 1216.05135
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