On graph irregularity strength

Abstract

An assignment of positive integer weights to the edges of a simple graph G is called irregular, if the weighted degrees of the vertices are all different. The irregularity strength, s(G), is the maximal weight, minimized over all irregular assignments. In this study, we show that s(G) c 1 n /, for graphs with maximum degree Á n 1=2 and minimum ------------------

Irregularity strength of dense graphs

“…, 2∆}, the number of vertices of G which have weights in I 1 is not too large. The value of l will be proportional to log n. The existence of such an assignment f is proved in Lemma 4 by using the same random assignment as in the proof of Lemma 9 of [15]. Clearly, this f could be very far from being irregular.…”

“…Since then not much progress was made towards understanding the irregularity strength of regular graphs until Frieze, Gould, Karoński, and Pfender provided several impressive new upper bounds on s(G) in [15]. In particular, their results offer further support to the affirmative answer to Q.…”

“…Our approach follows closely the one in [15], especially the proof of their Lemma 9. The main difference is that for a random assignment, we bound the probability of many vertices having their weights within a certain interval, rather than having a given weight, as was done in [15].…”

“…For many results which were not mentioned in the survey and appeared since it was published, see [1], [11], [16], [12], [10], [2], [17], [21], [20], [26], [3], [18], [23], [25], [15], [5], [4].…”

“…The main difference is that for a random assignment, we bound the probability of many vertices having their weights within a certain interval, rather than having a given weight, as was done in [15]. Another main difference is the way we modify a "good" random assignment into an irregular one: instead of using a factor of G consisting of generalized stars, we use a special collection of subgraphs, all isomorphic to K 2,a for some a = a(n).…”