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Irregularity Strength

Consider a simple graph with no isolated edges and at most one isolated point. Assign edge a positive integer weight , and define the corresponding vertex labels to be equal to the sum of weights of edges incident on . Then the irregularity strength of a graph (also denoted , e.g., Dinitz et al. 1992), is the smallest value of such that the vertex weights are all distinct (Chartrand et al. 1988, Przybylo 2024).

Define for a graph with one or more isolated edges or more than one isolated point.

For any graph other than the triangle graph on vertices with finite irregularity strength,

 (1)

(Nierhoff 2000, Przybylo 2024), with equality for star graphs with (Przybylo 2024).

For a connected graph on vertices, Chartrand et al. (1988) gave the lower bound

 (2)

where is the number of vertices of vertex degree and is the maximum vertex degree of . For a regular graph of degree , this condition reduces to

 (3)

(Przybylo 2024). Another bound attributed to Jacobson and Lehel defines

 (4)

then

 (5)

(Ebert et al. 1990, Dinitz et al. 1992, Bohman and Kravitz 2004). Note that some authors (e.g., Ebert et al. 1990, Bohman and Kravtiz 2004) leave as the potentially rational number defined above, while others (e.g., Dinitz et al. 1992) explicitly add a ceiling function around the right hand side to make it an integer.

For a graph with minimum vertex degree ,

 (6)

(Kalkowski et al. 2011, Przybylo 2024).

Faudree and Lehel (1987) conjectured that there exists an absolute constant such that for every -regular graph with vertices and ,

 (7)

(Przybylo 2024).

Irregular Graph, Regular Graph

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References

Amar, D. "Irregularity Strength of Regular Graphs of Large Degree." Disc. Math. 114, 9-17, 1993.Anholcer, M. and Palmer, C. "Irregular Labellings of Circulant Graphs." Discr. Math. 312, 3461-3466, 2012.Bača, M.; Imran, M.; and Semaničová-Feňovč’ková', A. "Irregularity and Modular Irregularity Strength of Wheels." Mathematics 9, 2710, 2021.Bohman, T. and Kravitz, D. "On the Irregularity Strength of Trees." J. Graph Th. 45, 241-254, 2004.Chartrand, G.; Jacobson, M. S.; Lehel, J.; Oellermann, O. R.; Ruiz, S.; and Saba, F. "Irregular Networks." Congr. Numer. 64, 197-210, 1988.Dinitz, J. H.; Garnick, D. K.; and Gyárfás, A. "On the Irregularity Strength of the Grid." J. Graph Th. 16, 355-374, 1992.Ebert, G., Hemmeter, J.; Lazebnik F.; and Woldar, A. "On the Number of Irregular Assignments on a Graph." Disc. Math. 93, 141-142, 1991.Ebert, G.; Hammenter, J.; Lazebnik, F.; and Woldar, A. "Irregularity Strengths for Certain Graphs." Congr. Numer. 71, 39-52, 1990.Faudree, R.; Schelp, R.; Jacobson, M.; and Lehel, J. "Irregular Networks, Regular Graphs and Integer Matrices With Distinct Row and Column Sums." Disc. Math. 76, 223-240, 1989.Faudree, R. J. and Lehel, J. "Bound on the Irregularity Strength of Regular Graphs." In Colloq. Math. Soc. Janós Bolyai, Vol. 52, Combinatorics, Eger. Amsterdam, Netherlands: North-Holland, pp. 247-256, 1987.Gallian, J. §7.13 in "Dynamic Survey of Graph Labeling." Elec. J. Combin. DS6. Dec. 21, 2018. https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6.Gyarfas, A. "The Irregularity Strength of is 4 for Odd." Discr. Math. 71, 273-274, 1988.Jacobson, M. S. and Lehel, J. "A Bound for the Strength of an Irregular Network." Preprint.Jendroľ, S. and Tkác, M. "The Irregularity Strength of ." Disc. Math. 145, 301-305, 1995.Jendrol', S. and Žoldák, V. "The Irregularity Strength of Generalized Petersen Graphs." Math. Slovaca 45, 107-113, 1995.Jinnah, M. I. and Santhosh Kumar, K. R. "Irregularity Strength of Triangular Snake and Double Triangular Snake." Adv. Appl. Disc. Math. 9, 83-92, 2012.Kalkowski, M.; Karoński, M.; and Pfender, F. "A New Upper Bound for the Irregularity Strength of Graphs." SIAM J. Disc. Math. 25, 1319-1321, 2011.Lehel, J. "Facts and Quests on Degree Irregular Assignments." In Graph Theory, Combinatorics and Applications: Proceedings of the Sixth Quadrennial International Conference on the Theory and Applications of Graphs, Western Michigan University (Ed. Y. Alavi, F. R. K. Chung, R. L. Graham, and D. F. Hsu). New York: Wiley, pp 765-782 1991.Nierhoff, T. "A Tight Bound on the Irregularity Strength of Graphs." SIAM J. Disc. Math. 13, 313-323, 2000.Przybylo, J. "The Irregularity Strength of Dense Graphs--On Asymptotically Optimal Solutions of Problems of Faudree, Jacobson, Kinch and Lehel." 13 Jun 2024. https://arxiv.org/abs/2406.09584.

Cite this as:

Weisstein, Eric W. "Irregularity Strength." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/IrregularityStrength.html