Irregularity Strength

Consider a simple graph with no isolated edges and at most one isolated point. Assign edge a positive integer weight 1<=w_(ij)<=k, and define the corresponding vertex labels w_i to be equal to the sum of weights of edges incident on v_i. Then the irregularity strength s(G) of a graph G (also denoted I(G), e.g., Dinitz et al. 1992), is the smallest value of k such that the vertex weights are all distinct (Chartrand et al. 1988, Przybylo 2024).

Define s(G)=infty for a graph with one or more isolated edges or more than one isolated point.

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


(Nierhoff 2000, Przybylo 2024), with equality for star graphs S_n with n>2 (Przybylo 2024).

For a connected graph on n>=3 vertices, Chartrand et al. (1988) gave the lower bound


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


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




(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 lambda(G) 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 delta>0,


(Kalkowski et al. 2011, Przybylo 2024).

Faudree and Lehel (1987) conjectured that there exists an absolute constant C such that for every d-regular graph G with n vertices and d>=2,


(Przybylo 2024).

See also

Irregular Graph, Regular Graph

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Cite this as:

Weisstein, Eric W. "Irregularity Strength." From MathWorld--A Wolfram Web Resource.