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

Given a simple graph $G$, the {\it irregularity strength} of $G$, denoted $s(G)$, is the least positive integer $k$ such that there is a weight assignment on edges $f: E(G) \to \{1,2,\dots, k\}$ for which each vertex weight $f^V(v):= \sum_{u: \{u,v\}\in E(G)} f(\{u,v\})$ is unique amongst all $v\in V(G)$. In 1987, Faudree and Lehel conjectured that there is a constant $c$ such that $s(G) \leq n/d + c$ for all $d$-regular graphs $G$ on $n$ vertices with $d>1$, whereas it is trivial that $s(G) \geq n/d$. In this short note we prove that the Faudree-Lehel Conjecture holds when $d \geq n^{0.8+\epsilon}$ for any fixed $\epsilon>0$, with a small additive constant $c=28$ for $d$ large enough. Furthermore, we confirm the conjecture asymptotically by proving that for any fixed $\beta\in(0,1/4)$ there is a constant $C$ such that for all $d$-regular graphs $G$, $s(G) \leq \frac{n}{d}(1+\frac{C}{d^{\beta}})+28$, extending and improving a recent result of Przyby{\l}o that $s(G) \leq \frac{n}{d}(1+ \frac{1}{\ln^{\epsilon/19}n})$ whenever $d\in [\ln^{1+\epsilon} n, n/\ln^{\epsilon}n]$ and $d$ is large enough.


Introduction
Let G be a simple graph with n vertices.For a positive integer k an edge-weighting function f : E(G) → {1, 2, . . ., k} is called k-irregular if the weighted degrees, denoted by f V (v) = u∈N (v) f ({v, u}) are distinct for v ∈ V (G); we will call f ({u, v}) and f V (v) simply the weights of {u, v} and v.The irregularity strength of G, denoted s(G), is the least k, if exists, for which there is such a k-irregular edge-weighting function f ; we set s(G) = ∞ otherwise.It is easy to see that s(G) < ∞ if and only if G has no isolated edges and at most one isolated vertex [7].
The irregularity strength was first introduced by Chartrand, Jacobson, Lehel, Oellermann, Ruiz, and Saba [3].Later an optimal general bound s(G) ≤ n − 1 was proved in [1,12] for all graphs with finite irregularity strength except for K 3 .This bound occurred to be far from optimum for graphs with larger minimum degree.Special concern was in this context devoted to d-regular graphs.In [6] Faudree and Lehel showed s(G) ≤ ⌈n/2⌉ + 9 for these.By a simple counting argument, it is easy to see that on the other hand, s(G) ≥ ⌈(n + d + 1)/d⌉.This lower bound motivated Faudree and Lehel to conjecture that (n/d) is close to optimal, as proposed in [6] in 1987.In fact this conjecture was first posed by Jacobson, as mentioned in [10].

Conjecture 1 ([6]
).There is a constant C > 0 such that for all d-regular graphs G on n vertices and with d > 1, s(G) It is this conjecture that "energized the study of the irregularity strength", as stated in [4], and many related subjects throughout the following decades.It remains open after more than thirty years since its formulation.A significant step forward towards solving it was achieved in 2002 by Frieze, Gould, Karoński, and Pfender, who used the probabilistic method to prove the first linear bound s(G) ≤ 48(n/d) + 1 for d ≤ √ n, and a super-linear one s(G) ≤ 240(log n)(n/d) + 1 in the remaining cases.The linear bound in n/d was further extended to the case when d ≥ 10 4/3 n 2/3 log 1/3 n by Cuckler and Lazebnik [4].The first general and unified linear bound in n/d for the full spectrum of (n, d) was delivered by Przyby lo [13,14], who used a constructive rather than random approach to prove the bound s(G) ≤ 16(n/d) + 6.Since then several works based on inventive new algorithms have been conducted to improve the multiplicative constant in front of n/d, see e.g.[8,9,11].The best result among these for any value of d is due to Kalkowski, Karoński, and Pfender [9], who showed that in general s(G) ≤ 6⌈n/δ⌉ for graphs with minimum degree δ ≥ 1 and without isolated edges.Only just recently it was proved by Przyby lo [15] that the Faudree-Lehel Conjecture holds asymptotically almost surely for random graphs G(n, p) (which are typically "close to" regular graphs), for any constant p, and holds asymptotically (in terms of d and n) for d not in extreme values.
Theorem 2 (Przyby lo [15]).Given any ǫ > 0, for every d-regular graph G with n vertices and 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.[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.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.

Preliminaries
For a set U ⊂ V (G) and a vertex v ∈ V (G), we use deg U (v) to denote the number of neighbors of v in U .For a positive constant x, let {x} stand for x − ⌊x⌋.We will use the following tools.
Lemma 7 (Chernoff Bound).Let X 1 , . . ., X n be i.i.d.random variables such that Pr(X i = 1) = p and Pr(X i = 0) = 1 − p for each i.Then for any t ≥ 0, Pr n i=1 Lemma 8 (Lovász Local Lemma).[5,2] Let E 1 , . . ., E n be n events in any given probability space.Let H be a simple graph with vertex set [n] such that for each i ∈ [n], the event E i is mutually independent from the remaining events corresponding to non-neighbors of the vertex i, i.e., {E j : j = i, {i, j} / ∈ E(H)}.Suppose there exist values x 1 , . . ., x n ∈ (0, 1) such that for each i ∈ [n], Pr Then the probability that none of the events E i happens is positive, i.e., Pr( n i=1 Ēi ) > 0.

Random vertex partition through local lemma
Some part of our construction builds on ideas from [15].In order to bypass the log n barrier for d and be able to analyze the algorithm for all 1 ≤ d ≤ n, we however need to phrase our construction differently, using quantization and the Lovász Local Lemma (Lemma 8).
The idea is to partition V (G) into a big set B = {v 1 , . . ., v |B| } and a small set S, where |S| = (n/d) • o(d).At the end, we will assure that f V (v i+1 ) = f V (v i ) + 1 in B, and that vertices in S have larger weights than those in B. Our argument divides into three steps.Step 1 includes a random construction positioning weights in B close to expected values, which are relatively sparsely distributed.In Step 2 we modify the weights of edges across B and S to make vertices in B have the desired weights.This is also the main purpose of singling out the set S. One benefit of S being small compared to B is that if we assign heavy weights between S and B, then weights of vertices in S are expected to increase more significantly than those in B. Step 3 is to modify weights in S in order to make them all pairwise distinct.Fix parameters ǫ, γ such that ǫ ∈ (0, 1/4) and 0 < 2γ < ǫ.Let G be an n-vertex d-regular graph.Set s * = 13⌈d 1/2+ǫ /13⌉, note that s * ∈ [d 1/2+ǫ , d 1/2+ǫ + 13) and 13|s * .Unless specified, we always assume d is sufficiently large in terms of γ.
We first describe the main random ingredient of the construction.Let X v for v ∈ V (G) be i.i.d.uniform random variables, X v ∼ U [0, 1].We use the values of X v 's to separate the vertices into d bins B i where note that in expectation, each B i includes n/d vertices.Let the big set, consisting of most of the bins be defined as B = 1≤i≤d−s * B i .The remaining bins form a small set S, which we partition into regular 13 subsets Finally, we label some edges as "corrected" to satisfy a subtle technical issue (and guarantee later that the average weight of edges weighted ⌊n/d⌋ + 1 and ⌊n/d⌋ + 2 is exactly (n/d) + 1).More precisely, we randomly label an edge with both end vertices in B "corrected" independently with probability max({n/d}, 1 − {n/d}), which is at least 1/2.Lemma 9.With positive probability, the following statements hold simultaneously if d is large enough.

(C
Proof.Let E vS i be the bad event that C vS i does not hold for given v ∈ V (G), 1 ≤ i ≤ 13.We analogously denote by E vS , E vB , E ′ vB , E i , E S i the remaining bad events.We first bound the probability of each of these, and then use Lovász Local Lemma to show that with positive probability none of these bad events happen.
Fix v ∈ V (G) and let us consider E vS i for any given 1 ≤ i ≤ 13.Since each of d neighbors of v is independently included in S i with probability exactly s * /(13d), where d 1/2+γ < s * /13 < d for d large enough, by the Chernoff Bound,

As the events C vS
Since by definition Pr(E vB |v ∈ S) = 0 and Pr(E ′ vB |v ∈ S) = 0, thus by the law of total probability, Pr(E vB ) < 2e −d 2γ /6 and Pr(E ′ vB ) < 2e −d 2γ /6 .To finally estimate Pr(E i ), we note that for any i each of n vertices is independently included in j≤i B i with probability i/d.Thus by the Chernoff Bound, Since conditions ( 1) and ( 5) of the lemma imply conditions ( 2) and ( 6), respectively, we just need to show that with positive probability none of E vS i , E vB , E ′ vB , E i holds.We will apply the Lovász Local Lemma (Lemma 8).There are 13n events of type E vS i (for each v ∈ V (G) and 1 ≤ i ≤ 13), n events of type E vB , n events of type E ′ vB and d events of type E i .Note that for any given v and i, each of the events E vS i , E vB , E ′ vB is mutually independent of all other events E uS j , E uB , E ′ uB with u at distance at least 3 from v in G, i.e. all but most 13(d 2 + 1) + (d 2 + 1) + (d 2 + 1) < 16d 2 such events.We assign value x = d −2 /1600 to each E vS i , E vB , E ′ vB , and assign value y = d −1 /100 to all E i .Therefore, in order to apply Lemma 8 we just need to check that Note that 1 − a ≥ e −10a for 0 ≤ a ≤ 0.5.Thus it is sufficient to show that: which is equivalent to: .
These two inequalities above hold when d is sufficiently large in terms of γ.The thesis thus follows by Lemma 8.

Assigning weights
Suppose all statements in Lemma 9 hold.We will assign and modify edge weights in G in three steps.Whenever needed we assume d is large enough in terms of γ.
Step 1.The purpose of this step is to construct an initial weighting function f 1 : E(G) → N so that all v ∈ B i have weights very close to (n/d)i for each i.For this aim for every edge {u, v} with v ∈ B i ∩ B and u ∈ B j ∩ B we define f 1 ({u, v}) as follows.If {n/d} ≥ 1/2 and d − s v} is not a corrected edge, and let f 1 ({u, v}) = ⌊n/d⌋ + 1 if it is a corrected edge.We next define ω = max(⌈n/d 1+ǫ−2γ ⌉, 2) and set f 1 (e) = iω + ⌈n/d⌉ for every edge e across B and S i for 1 ≤ i ≤ 13.We finally set f 1 (e ′ ) = 1 for all the remaining edges e ′ ∈ E(G).Consider any v ∈ B i ∩B.We assume {n/d} ≥ 1/2, as the analysis and result in the opposite case is essentially the same.By the definition of f 1 and Lemma 9, since 1 < n/d and ω ≤ ⌈n/d⌉ < 2n/d, By almost the same reasoning we may obtain an analogous upper bound for f set in AP that contains the current value of f V (v 1 ) and move on (to v 2 ).Then for every consecutive i ≥ 2, we will choose a special set AP v i ∈ AP and guarantee that f V (v i ) belongs in AP v i since the end of step i till the end of entire algorithm.We admit two options to modify f 3 on backward edges of the given v i : either by adding 0 or one of the values in {±⌈n/(3d)⌉}.Specifically, say {v i , u} is a backward edge of v i .If the current value of f V (u) is the smaller value in AP u , we admit adding 0 or ⌈n/(3d)⌉ to f 3 ({v i , u}); if f V (u) is the larger value in AP u , we in turn admit subtracting 0 or ⌈n/(3d)⌉ from f 3 ({v i , u}).Thereby the updated f V (u) will always remain in AP u , as desired.We finally admit adding any value in {0, 1, . . ., ⌈3n/d⌉} to the weights f 3 (e) of all forward edges of v i , which will in particular allow us to determine the congruence class f V (v i ) will eventually land in.We now specify the ordering v 1 , v 2 , . . . of the vertices in S. At the beginning we arrange the vertices in 1≤i≤12 S i according to the values of X v i , from the smallest to the largest, and thus consistently with the order of S 1 , . . ., S 12 .The last in the ordering are vertices from S 13 , which are ordered differently due to some technical subtlety concerning vertices without forward edges.Suppose C 1 , . . ., C K are the connected components in S 13 , ordered arbitrarily.Each component has at least two vertices by Lemma 9(1).For each C i , we use reversed BFS to order its vertices and denote r i , t i the last two vertices in C i .(Thus t i is the root of the tree in BFS; {r i , t i } ∈ E(S).)Let R = {r 1 , . . ., r K } and T = {t 1 , . . ., t K }.We finally define the ordering in S 13 by concatenating the orderings of C 1 , . . ., C K .Note that by Lemma 9(1), the set of terminal vertices, i.e., vertices with no forward edges in the obtained ordering in S, is T .
We now show specific procedures which will allows us to achieve the desired goal.Suppose we are in step i, i.e. we are analyzing v i ∈ S t , where 1 ≤ t ≤ 13, and that v i / ∈ R ∪ T , hence v i has at least one forward edge, say e i .The existing sets AP u for u prior to v i in S t correspond to at most |S t | congruence classes with possible duplicates.Therefore, there must be a congruence class C a that includes at most |S t |/⌈n/(3d)⌉ prior sets AP u with u ∈ S t .Thus we may include the weight of f V (v i ) in C a by adding one of admissible values in {0, 1, . . ., ⌈3n/d⌉} to the weight f 3 (e i ).We then modify the rest of the forward edges of v i by adding 0 or ⌈n/(3d)⌉ and change the weights of some backward edges of v i by ⌈n/(3d)⌉ according to the specified rules, if necessary.Note that this way we may obtain deg S (v i ) consecutive terms in C a as weights of v i .Since each prior set AP u blocks at most two consecutive terms in C a , we can find this way an attainable f V (v i ) ∈ C a which is not blocked if only deg S (v i ) > 2|S t |/⌈n/(3d)⌉.This is however implied by an even stronger inequality, which holds by Lemma 9 (6)(2): 4|S t |/⌈n/(3d)⌉ + 2 ≤4(s * n/(13d) + 2n/d 1/2−γ )/(n/(3d)) + 2 ≤ 12s * /13 + 24d 1/2+γ + 2 We finally set AP v i as the only set in AP containing the attained weight of v i .We are left to show how to handle r j , t j ∈ R ∪ T , where {r j , t j } is the only forward edge of r j .We analyze both vertices simultaneously in a similar manner as above.Recall r j , t j ∈ S 13 .First, by an averaging argument, we can choose an admissible addition to f 3 ({r j , t j }) such that the two new congruence classes of f V (r j ), f V (t j ) each includes at most 2|S 13 |/⌈n/(3d)⌉ prior sets AP u with u ∈ S 13 , disregarding temporarily AP r j from the point of view of t j .Next, analogously as above, by (6), we can change the weights of backward edges of r j by ±⌈n/(3d)⌉ so that the resulting f V (r j ) belongs to AP r j ∈ AP disjoint from those of the prior vertices in S 13 .Finally, we analogously adjust the weights of all backward edges of t j except {r j , t j } so that the resulting f V (t j ) belongs to AP t j ∈ AP disjoint from those of the prior vertices in S 13 including AP r j , which is again feasible by (6) (where "+2" was incorporated in this inequality to facilitate distinguishing AP t j from AP r j ).Claim 14.For every edge e of G, 1 ≤ f (e) ≤ ⌈n/d⌉ + 13ω + ⌈10 3 n/d 1+ǫ−γ ⌉.
i imply C vS , we proceed to compute the conditional probabilities Pr(E vB |v ∈ B i ) and Pr(E ′ vB |v ∈ B i ).These are trivially 0 for i = 1.Thus we next assume 2 ≤ i ≤ d − s * .As each of d neighbors u of v has probability (i − 1)/d to be in d−s * −i+1<j≤d−s * B j and probability α(i − 1)/d to be in d−s * −i+1<j≤d−s * B j and simultaneously form a corrected edge uv, by the two Chernoff Bounds, since α ≥ 1/2, 1/19).Given any 0 < β < 1/4, there is a constant C such that for every d-regular graph G on n vertices with d 1+β ≥ n, s(G) < n/d + C.