# Group Irregular Labelings of Disconnected Graphs

Keywords:
Irregularity strength, Graph weighting, Graph labeling, Abelian group

### Abstract

We investigate the \textit{group irregularity strength} ($s_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group $\gr$ of order $s$, there exists a function $f:E(G)\rightarrow \gr$ such that the sums of edge labels at every vertex are distinct. We give the exact values and bounds on $s_g(G)$ for chosen families of disconnected graphs. In addition we present some results for the \textit{modular edge gracefulness} $k(G)$, i.e. the smallest value of $s$ such that there exists a function $f:E(G)\rightarrow \zet_s$ such that the sums of edge labels at every vertex are distinct.### References

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M. Kalkowski, M. Karonski, and F. Pfender, A new upper bound for the irregularity strength of graphs, SIAM J. Discrete Math. 25 (2011), 1319-1321.

G. Kaplan, A. Lev, and Y. Roditty, On zero-sum partitions and anti-magic trees, Discrete Math. 309 (2009), 2010-2014.

J. Lehel, Facts and quests on degree irregular assignments, Graph Theory, combinatorics and applications : proc. of the Sixth Quadrennial International Conference on the Theory and Applications of Graphs, Wiley, New York, 1991, pp. 765-782.

S. P. Lo, On edge-graceful labelings of graphs, Congr. Numer. 50 (1985), 231-241.

P. Majerski and J. Przyby lo, On the irregularity strength of dense graphs, SIAM J. Discrete Math. 28 (2014), no. 1, 197-205.

O. Togni, Force des graphes. indice optique des reseaux, Ph.D. thesis, Universite de Bordeaux 1, Ecole doctorale de mathematiques et d'informatique, 1998.

D. Amar and O. Togni, Irregularity strength of trees, Discrete Math. 190 (1998), 15-38.

M. Anholcer, S. Cichacz, and M. Milanic, Group irregularity strength of connected graphs, J. Comb. Optim. 30 (2015), 1-17.

G. Chartrand, M. S. Jacobson, J. Lehel, O. R. Oellermann, S. Ruiz, and F. Saba, Irregular networks, Congr. Numer. 64 (1988), 187-192.

M. Ferrara, R. J. Gould, M. Karonski, and F. Pfender, An iterative approach to graph irregularity strength, Discrete Appl. Math. 158 (2010), 1189-1194.

F. Fujie-Okamoto, R. Jones, K. Kolasinski, and P. Zhang, On modular edge-graceful graphs, Graphs Combin. 29 (2013), 901-912.

J. Gallian, Contemporary abstract algebra, seventh ed., Brooks/Cole Cengage Learning, 2010.

R. Jones, Modular and graceful edge colorings of graphs, Ph.D. thesis, Western Michigan University, 2011.

R. Jones and P. Zhang, Nowhere-zero modular edge-graceful graphs, Discuss. Math. Graph Theory 32 (2012), 487-505.

M. Kalkowski, M. Karonski, and F. Pfender, A new upper bound for the irregularity strength of graphs, SIAM J. Discrete Math. 25 (2011), 1319-1321.

G. Kaplan, A. Lev, and Y. Roditty, On zero-sum partitions and anti-magic trees, Discrete Math. 309 (2009), 2010-2014.

J. Lehel, Facts and quests on degree irregular assignments, Graph Theory, combinatorics and applications : proc. of the Sixth Quadrennial International Conference on the Theory and Applications of Graphs, Wiley, New York, 1991, pp. 765-782.

S. P. Lo, On edge-graceful labelings of graphs, Congr. Numer. 50 (1985), 231-241.

P. Majerski and J. Przyby lo, On the irregularity strength of dense graphs, SIAM J. Discrete Math. 28 (2014), no. 1, 197-205.

O. Togni, Force des graphes. indice optique des reseaux, Ph.D. thesis, Universite de Bordeaux 1, Ecole doctorale de mathematiques et d'informatique, 1998.

Published

2017-11-27

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