On the asymptotic confirmation of the Faudree-Lehel Conjecture for general graphs

Abstract

Given a simple graph G, the irregularity strength of G, denoted by s(G), is the least positive integer k such that there is a weight assignment on edges f : E(G) → {1, 2, . . . , k} attributing distinct weighted degrees: f (v) := u:{u,v}∈E(G) f ({u, v}) to all vertices v ∈ V (G). It is straightforward that s(G) ≥ n/d for every d-regular graph G on n vertices with d > 1. In 1987, Faudree and Lehel conjectured in turn that there is an absolute constant c such that s(G) ≤ n/d + c for all such graphs. Even though the conjecture has remained open in almost all relevant cases, it is more generally believed that there exists a universal constant c such that s(G) ≤ n/δ+c for every graph G on n vertices with minimum degree δ ≥ 1 which does not contain an isolated edge. In this paper we confirm that the generalized Faudree-Lehel Conjecture holds for graphs with δ ≥ n β where β is any fixed constant larger than 0.8. Furthermore, we confirm that the conjecture holds in general asymptotically. That is we prove that for any ε ∈ (0, 0.25) there exist absolute constants c 1 , c 2 such that for all graphs G on n vertices with minimum degree δ ≥ 1 and without isolated edges, s(G)thus extending in various aspects and strengthening a recent result of Przyby lo, who showed that s(G) ≤ n d (1 + 1 ln /19 n ) = n d (1 + o(1)) for d-regular graphs with d ∈ [ln 1+ n, n/ ln n], and improving an earlier general upper bound: s(G) < 6 n δ + 6 of Kalkowski, Karoński and Pfender.

Short proof of the asymptotic confirmation of the Faudree-Lehel Conjecture

“…In [15], Przyby lo moreover mentioned that "a poly-logarithmic in n lower bound on d is unfortunately unavoidable" within his approach. In this paper we present an argument which is firstly quite short, secondly bypasses the mentioned poly-logarithmic in n lower bound and extends the asymptotic bound to all possible cases 1 ≤ d ≤ n − 1 and thirdly, the upper bound we present is stronger than the one in Theorem 2 (where in particular ln ǫ/19 n ≪ ln (1+ǫ)/19 n ≤ d We remark that similar conclusions as the ones above can also be derived from [16], which describes on almost 30 pages a very long, multistage and technically complex random construction yielding general results for all graphs (not only regular graphs). Taking into account that Conjecture 1 remains a central open question of the related field, cf.…”

“…Taking into account that Conjecture 1 remains a central open question of the related field, cf. [16] for more comprehensive exposition of the history and relevance of this problem, we decided to present separately this very concise argument concerning the conjecture itself, which is also dramatically easier to follow. Moreover, the present proof is a local lemma based argument, and thus is very different from the one in [16], which might also be beneficial for further research.…”

“…[16] for more comprehensive exposition of the history and relevance of this problem, we decided to present separately this very concise argument concerning the conjecture itself, which is also dramatically easier to follow. Moreover, the present proof is a local lemma based argument, and thus is very different from the one in [16], which might also be beneficial for further research. Another note is that the current proof is somewhat using the full power of the local lemma, where the symmetric version of local lemma is insufficient.…”

“…Another note is that the current proof is somewhat using the full power of the local lemma, where the symmetric version of local lemma is insufficient. Lastly, unlike in [16], we also provide a specific additive constant in the obtained bounds for regular graphs, in particular in Theorem 5, which is relatively small.…”

Irregular subgraphs

“…In Step 3, we will only adjust the weights of edges within S to ensure that each weight appears in at most 2016n ln n ln ln n δ 1+ + 1 vertices in S. We first use a method developed in a paper in preparation by the second author and J. Przybyło [16] to identify which vertices in S might have the same weight. For each vertex v ∈ S, we will define a set L(v) such that v, u ∈ S cannot have the same weight at the end of Step 3 if u / ∈ L(v).…”