(V,E) be a hypergraph on t - Semantic Scholar

THE 1-2-3 CONJECTURE FOR UNIFORM HYPERGRAPHS PATRICK BENNETT, ANDRZEJ DUDEK, ALAN FRIEZE, LAARS HELENIUS November 14, 2015 Abstract. Given an r-uniform hypergraph H = (V, E) and a weight function ω : E P → {1, . . . , w}, a coloring of vertices of H, induced by ω, is defined by c(v) = e3v w(e) for all v ∈ V . In this paper, we show that for almost all 3-uniform hypergraphs there exists a weighting of the edges from {1, 2} that induces a proper vertex-coloring (that means with no monochromatic edges). For r ≥ 4, we show that almost all r-uniform hypergraphs can be weighted just from {1}. These results extend a previous work of Addario-Berry, Dalal and Reed for graphs. Finally, we also give a sufficient condition for uniform hypergraphs to have a proper coloring induced by a weighting from {1, 2}.

1. Introduction We start with a few definitions. Let H = (V, E) be a hypergraph on the vertex set V and with the set of edges E ⊆ 2V . Let ω : E → {1, . . . , w} be a weight function. Furthermore, let c : V → N be a vertex-coloring induced by ω defined as X ω(e) c(v) = v3e

for each v ∈ V . The vertex-coloring is proper if no edge is monochromatic. We say that H is w-weighted if there exists ω : E → {1, . . . , w} such that the vertex-coloring c induced by ω is proper. In this paper we mainly deal with r-uniform hypergraphs. A hypergraph H = (V, E) is r-uniform if for each e ∈ E, |e| = r. (Clearly, a 2-uniform hypergraph is just a graph.) In 2004 Karo´ nski, Luczak and Thomason [10] proposed the following conjecture. 1-2-3 Conjecture ([10]). Every graph without isolated edges is 3-weighted. This conjecture attracted a lot of attention and has been studied by several researchers (see, e.g., a survey paper of Seamone [11]). The 1-2-3 conjecture is still open but it is known due to a result of Kalkowski, Karo´ nski, and Pfender [8] that every graph without isolated edges is 5-weighted. Quite recently, the same authors [9] started to study analogous problems for hypergraphs. In particular, they proved that any r-uniform hypergraph is (5r − 5)-weighted The second author was sponsored by the National Security Agency under Grant Number H9823015-1-0172. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation hereon. The third author was supported in part by NSF grant DMS1362785. 1

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PATRICK BENNETT, ANDRZEJ DUDEK, ALAN FRIEZE, LAARS HELENIUS

f1 e1

e2

f3

f2

e3

Figure 1. A 4-uniform hypergraph which is not 2-weighted. and that 3-uniform hypergraphs are even 5-weighted. The authors also asked whether there is an absolute constant w0 such that every r-uniform hypergraph is w0 -weighted. Furthermore, they conjectured that each 3-uniform hypergraph without isolated edges is 3-weighted. In this paper we show that for almost all uniform hypergraphs these conjectures hold. We say that almost all r-uniform hypergraphs on n vertices have property P if as n n
tends to infinity, o 2( r ) r-uniform hypergraphs on n vertices do not have property P. Theorem 1.1. (i) Almost all 3-uniform hypergraphs are 2-weighted (but not 1-weighted). (ii) For a fixed r ≥ 4 almost all r-uniform hypergraphs are 1-weighted. The second part is easier, since a hypergraph is 1-weighted if and only if there is no edge consisting of vertices that all have the same degree. Thus, the interesting part is showing that almost all 3-uniform hypergraphs are 2-weighted. This theorem (proven in Sections 3 and 4) extends a result of Addario-Berry, Dalal and Reed [1] who showed that almost all graphs are 2-weighted. However, our proofs our completely different. It is not difficult to see that in general r-uniform hypergraphs are not 2-weighted. Indeed, define an r-uniform hypergraph H = (V, E) on V = {x1 , . . . , xr , y1 , . . . , yr , z1 , . . . , zr } with edges e1 = {x1 , . . . , xr }, e2 = {y1 , . . . , yr }, e3 = {z1 , . . . , zr }, f1 = {x2 , . . . , xr , y1 }, f2 = {y2 , . . . , yr , z1 }, and f3 = {z2 , . . . , zr , x1 } (see Figure 1). If there is a proper coloring of H induced by some w : E → {1, 2}, then for some i, j we must have ω(ei ) = ω(ej ) and consequently edge f` ⊆ V (ei ) ∪ V (ej ) is monochromatic, a contradiction. We believe that all uniform hypergraphs without isolated edges are 3-weighted. As a matter of fact we will show that a large class of hypergraphs is 2-weighted. An example of such class consists of all r-uniform hypergraphs for r ≥ 5 and minimum degree Ω(n4 ) and pair degree “not too big”(see Section 5). 2. Preliminaries In order to prove Theorem 1.1 we will consider random
hypergraphs. Let H(k) (n, p) be an r-uniform random hypergraph such that each of [n] r-tuples is contained with r

THE 1-2-3 CONJECTURE FOR UNIFORM HYPERGRAPHS

3

probability p, independently of others. We say that an event En occurs with high probability, or w.h.p. for brevity, if limn→∞ Pr(En ) = 1. Hence, to show that almost all r-uniform hypergraphs on n vertices have property P as n tends to infinity it suffices to show that w.h.p. H(k) (n, 1/2) ∈ P. This paper uses some standard probabilistic tools, which we state here for convenience (for more details see, e.g., [2, 7]). First Moment Method. Let X be a nonnegative integral random variable. If E(X) = o(1), then w.h.p. X = 0. Second Moment Method. Let X be a nonnegative integral random variable. If Var(X) = o(E(X)2 ), then w.h.p. X ≥ 1. Let Bin(n, p) denotes the random variable with binomial distribution with number of trials n and probability of success p. Chernoff ’s Bound. If X ∼ Bin(n, p) and 0 < γ ≤ E(X), then
Pr (|X − E(X)| ≥ γ) ≤ 2 exp −γ 2 /(3E(X)) . Bernstein’s Bound. Let X1 , . . . , Xm be independent random variables, and Pm X = i=1 Xi . Suppose that |Xi − E(Xi )| ≤ C always holds for all i. Then, for all positive γ   1 2 γ 2 Pr (|X − E(X)| ≥ γ) ≤ 2 exp − Pm . 1 i=1 Var(Xj ) + 3 Cγ We will also use the union bound. Union Bound. If Ei are events, then m [  Pr Ei ≤ m · max{Pr(Ei ) : i ∈ [m]}. i=1

Several times we will also need to estimate binomial coefficients (for more details see, e.g., Chapter 22 in [6]). Binomial Coefficients Approximation. (A1) Let k > 0 and ` be functions of m (` can be negative). Assume that `2 = o(k) and `2 = o(m − k) as m tends to infinity. Then, `     m m m−k . ∼ k k+` k (A2) Let k ≥ 1 be a fixed integer. Then, !k r r m  k X m 2 πm ∼ 2m . i πm 2k i=0 (A3) 

m bm/2c



2m+1/2 ∼ √ . πm

All logarithms in this paper are natural (base e).

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PATRICK BENNETT, ANDRZEJ DUDEK, ALAN FRIEZE, LAARS HELENIUS

3. Almost all 3-uniform hypergraphs are 2-weighted First we show by using the second moment method that almost all 3-uniform hypergraphs are not 1-weighted. Recall that this is equivalent to showing the following: Proposition 3.1. Let H = H(3) (n, 1/2). Then, w.h.p. there is an edge {v1 , v2 , v3 } in H such that deg(v1 ) = deg(v2 ) = deg(v3 ). Proof. Let X be a random variable that counts the number of edges {v1 , v2 , v3 } in H(3) (n, 1/2) = ([n], E) such that deg(v1 ) = deg(v2 ) = deg(v3 ).
First we compute the expected value of X. Fix e = {v1 , v2 , v3 } ∈ [n] and con3 dition on e ∈ E. Let xi be the random variable that counts the number of edges containing only vi (and neither of the two remaining vertices). Furthermore, let yi,j be the random variable that counts the number
of edges containing vi and v
j (and
not [3] n−3 the remaining third vertex), where {i, j} ∈ 2 . Clearly, xi ∼ Bin , 1/2 and 2 yi,j ∼ Bin (n − 3, 1/2). The Chernoff bound together with the union bound yield that w.h.p. for any e, i, and j,   p p n − 3 xi − = O(n log n) and |yi,j − (n − 3)/2| = O( n log n). (1) /2 2 Clearly, for any e = {v1 , v2 , v3 } we have deg(v1 ) = x1 + y1,2 + y1,3 + 1, deg(v2 ) = x2 + y1,2 + y2,3 + 1, deg(v3 ) = x3 + y1,3 + y2,3 + 1, and so w.h.p. for any e and i   p deg(vi ) − n − 3 /2 = O(n log n). 2 Conditioning on y1,2 = β1,2 and y1,3 = β1,3 , we get that the probability that deg(v1 ) = a is
  n−3 n−3 2 2−( 2 ) . a − β1,2 − β1,3 − 1
√ /2 = O(n log n), where Due to (1) we may assume that βi,j = n2 + `i,j and a − n−3 2 √ |`i,j | = O( n log n).
By (A1) (applied with m = n−3 , k = a − n, and ` = `1,2 + `1,3 + 1) we obtain that 2

    n−3 n−3 2 2 = a − β1,2 − β1,3 − 1 (a − n) − (`1,2 + `1,3 + 1) !`1,2 +`1,3 +1
 n−3
 n−3 − (a − n) 2 2 ∼ . a−n a−n Now

n−3 2




− (a − n) ≤ a−n

n−3 /2 2
n−3 /2 2




√   + O(n log n) log n =1+O √ n − O(n log n)

THE 1-2-3 CONJECTURE FOR UNIFORM HYPERGRAPHS

and n−3 2

n−3 /2 2
n−3 /2 2







− (a − n) ≥ a−n

5

√   − O(n log n) log n =1−O . √ n + O(n log n)

√ Hence, since |`1,2 + `1,3 + 1| = O( n log n), we get that !`1,2 +`1,3 +1
n−3 − (a − n) 2 ∼ 1, a−n and consequently, n−3 2






 ∼

a − β1,2 − β1,3 − 1 Similarly, 

n−3 2






a − β1,2 − β2,3 − 1

n−3 2




 ∼

a−n

n−3 2


 . a−n



n−3 2




 and



a − β1,3 − β2,3 − 1

∼

n−3 2


 . a−n



Thus, conditioning on βi,j ’s and e ∈ E the probability that deg(v1 ) = deg(v2 ) = deg(v3 ) = a equals


    n−3 n−3 n−3 n−3 2 2 2 2−3( 2 ) a − β1,2 − β1,3 − 1 a − β1,2 − β2,3 − 1 a − β1,3 − β2,3 − 1  n−3
3 n−3 2 ∼ 2−3( 2 ) . a−n So, Pr(deg v1 = deg(v2 ) = deg(v3 ), {v1 , v2 , v3 } ∈ E)
X  n−3 3 n−3 1X 2 ∼ Pr(yi,j = βi,j ) 2−3( 2 ) , a−n 2 β a i,j

the
sums over all values of βi,j = wheren−3 are taken √ a − /2 = O(n log n). Since 2 X

Pr(yi,j = βi,j ) ∼

βi,j

n−3 X

n 2

√ + `i,j with |`i,j | = O( n log n) and

Pr(yi,j = βi,j ) = 1

βi,j =0

and X a


3

n−3 2

a−n

∼

n−3 (X 2 )

a=0


3

n−3 2

a

,

we get
(n−3 2 )  n−3 3 n−3 1 X 2 2 2−3( 2 ) ∼ √ , Pr(deg v1 = deg(v2 ) = deg(v3 ), {v1 , v2 , v3 } ∈ E) ∼ 2 a=0 a π 3n2

6

PATRICK BENNETT, ANDRZEJ DUDEK, ALAN FRIEZE, LAARS HELENIUS

where the latter follows from (A2) (applied with m =   n 2 √ , E(X) ∼ 3 π 3n2

n−3 2




and k = 3). Thus,

which goes to infinity together with n.
Now we compute the variance. We show that E(X 2 ) ∼ (E(X))2 . For e ∈ [n] , let 3 Xe be an indicator random variable which is equal to 1 if all three vertices in e have the same degree. Thus, X X X X2 = X + Xe Xf + Xe Xf + X e Xf . |e∩f |=0

|e∩f |=1

|e∩f |=2

If |e ∩ f | = 2, then we may assume that e ∪ f = {v1 , . . . , v4 } and so similarly to the previous computations Pr(Xe Xf = 1) = Pr(deg(v1 ) = · · · = deg(v4 ) and e, f ∈ E)
4    n−4 1 X n−4 1 −4 2 ∼ 2 ( 2 )=O 4 a n3 a and consequently, X |e∩f |=2



1 E(Xe Xf ) = O n · 3 n 4



= O(n) = o((E(X))2 ).

Similarly, 

 1 E(Xe Xf ) = O n · 4 = O(n) = o((E(X))2 ). n |e∩f |=1 P Finally, we describe how to compute |e∩f |=0 E(Xe Xf ). Fix two vertex disjoint triples
e = {v1 , v2 , v3 } and f = {v4 , v5 , v6 } in [n] and condition on e, f ∈ E. Similarly 3 as before we define xi and yi,j . Furthermore, we define zi,j,k that counts the number of edges containing 1 vertex

from e and 2 from f or 2 vertices from e and 1 from f . Clearly, xi ∼ Bin n−6 , 1/2 and yi,j ∼ Bin (n − 6, 1/2), and zi,j,k are indicator random 2 variables. Now for example, conditioning on e ∈ E we get X X X z1,j,k + 1. deg(v1 ) = x1 + y1,j + z1,j,k + X

2≤j≤6

5

4≤j n log n) ≤ 2 exp − = exp −Ω(log2 n) 3m/2 and so the union bound gives us that the probability there exists v, i, j such that |di,j (v) − m/2| > n log n is at most  3n · exp −Ω(log2 n) = o(1). Hence, w.h.p. for every vertex v and distinct parts Vi , Vj there are n2 /18 + O(n log n) edges containing v and additionally one vertex from each of Vi , Vj . Similarly (using the Chernoff and union bounds), w.h.p. for any vertex v and part Vi there are n2 /36 + O(n log n) edges containing v and additionally two vertices from Vi . For a while we will assume that n is divisible by 3. Now due to Lemma 3.3 there is w.h.p. a family M of disjoint perfect matchings where |M| = bγn2 c and each edge in each matching goes between the partition (i.e. has one vertex in each Vi ). Henceforth

THE 1-2-3 CONJECTURE FOR UNIFORM HYPERGRAPHS

9

we assume that all the edges of H were revealed obtaining a hypergraph H that has the degree properties mentioned in the previous paragraphs, and the family of matchings M. Next we reveal the vertex labels and the non-matching edge weights. The weight of edge {x, y, z} is distributed as ( 1 with probability 1 − `(x) − `(y) − `(z) w(x, y, z) = 2 with probability `(x) + `(y) + `(z) so E(w(x, y, z) | `(x), `(y), `(z)) = 1 + `(x) + `(y) + `(z) and therefore if by cH\M (x) we denote the sum of the weights of non-matching edges containing x, and say we are given only the label `(x), and the hypergraph H with family of matchings M, and x ∈ Vi then we have X E(cH\M (x) | `(x), H) = E(w(x, v, w) | `(x)) {x,v,w}∈E(H)\M

X

=

1 + `(x) + E(`(v)) + E(`(w))

{x,v,w}∈E(H)\M

X

= (1 + `(x)) degH\M (x) +

E(`(v)) + E(`(w))

{x,v,w}∈E(H)\M

 = `(x)

 1 − γ n2 + Θ(n2 ), 4

where on the last line we have used our estimate of degH\M (x). Note that the Θ(n2 ) term may depend on i (recall x ∈ Vi ) but does not otherwise depend on x or `(x) (by the fact that the degree ofP each vertex v into sets Vi , Vj etc. is concentrated). Now since cH\M (x) = H\M w(x, v, w), and the random variables w(x, v, w) are independent (given the vertex labels), we can apply Bernstein’s inequality. For our application we use m = degH\M (x). We can easily use C = 2, and put Var(w(x, v, w)) ≤ E(w(x, v, w)2 ) ≤ 4. We will set γ = n log n. Then we get log2 n 4 degH\M (x) + 13 · 2n log n 
= exp −Ω log2 n



Pr cH\M (x) − E cH\M (x) | `(x), H > n log n ≤ 2 exp −

1 2 n 2

and so by the union bound, w.h.p. for each vertex x, cH\M (x) is within n log n of its expectation. Now for any fixed integer a and fixed vertex x,     1 2 P r `(x) − γ n − a ≤ n log n = O(log n/n), 4 since this is the probability that `(x) falls within an interval of length O(log n/n). Since the vertex labels
are independent, the probability that there are log2 n many vertices x with `(x) 14 − γ n2 − a ≤ n log n is at most

!

10

PATRICK BENNETT, ANDRZEJ DUDEK, ALAN FRIEZE, LAARS HELENIUS

  log2 n 2 n ne log2 n · (O(log n/n)) ≤ (O(log n/n))log n 2 2 log n log n   log2 n 1 = O = o(n2 ). log n



Therefore, by the union bound over integers c from 0 to n2 , we have that for all such c the number of vertices x with |cH\M (x) − c| ≤ n log n is at most log2 n. Henceforth we assume that the labels and non-matching edge weights have been revealed and they satisfy this property. Now we reveal the matching edge weights. These are 1 or 2 with probability 1/2. We will denote by cM (x) the sum of the weights of matching edges adjacent to x; note that c(x) = cM (x)+cH\M (x). The key fact we will use here is that since |M| = bγn2 c, cM (x) is not likely to be any one particular value. Indeed, since cM (x)−|M| ∼ Bin(|M|, 1/2), the mode of cM (x) occurs with probability p   2/γπ |M| −|M| 2 ∼ = O(1/n), b|M|/2c n where the latter follows from (A3). Also, w.h.p. for all x we have |cM (x) − 32 γn2 | ≤ n log n. Therefore, 2

n X


E (|{{x, y, z} : c(x) = c(y) = c(z) = c}|) ≤ o(1) + O n2 · (log2 n)3 · (1/n)3 = o(1)

c=0

on the last line we get the (1/n)3 in the big-O term since the probability that c(x) = c is O(1/n), and conditioning on that event, the event that c(y) = c still has probability O(1/n) (since revealing c(x) only reveals the weights of O(n) of the bγn2 c matching edges containing y, c(y) still essentially has the same distribution as it did before conditioning) and similarly for the conditional probability that c(z) = c. If n is not divisible by 3, then we can still use an equipartition V1 , V2 , V3 , and a family of (not quite perfect) matchings M where |M| = γn2 , each M ∈ M has size bn/3c, and each vertex is covered by γn2 − O(n) matchings in M.  4. Almost all r-uniform hypergraphs with r ≥ 4 are 1-weighted Proposition 4.1. Let r ≥ 4 and H = H(r) (n, 1/2). Then, w.h.p. H contains no four vertices v1 , v2 , v3 , and v4 such that deg(v1 ) = · · · = deg(v4 ). Proof. Let X be the random variable that counts the number of quadruples {v1 , v2 , v3 , v4 } in H such that deg(v1 ) = · · · = deg(v4 ). Fix v1 , v2 , v3 , and v4 . Let xi be the random variable that counts the number of edges containing vi and no other vertex from {v1 , . . . , v4 }, where i = 1, 2, 3, 4. Let yi,j be the random variable that counts the number
of edges [4] containing vi and vj (and no other remaining vertices), where {i, j} ∈ 2 . Similarly we define zi,j,k as the random variable that counts number of edges containing vi , vj and

THE 1-2-3 CONJECTURE FOR UNIFORM HYPERGRAPHS

vk , where {i, j, k} ∈

[4] 3

11


. Observe that if deg(v1 ) = · · · = deg(v4 ), then

x1 + y1,2 + y1,3 + y1,4 + z1,2,3 + z1,2,4 + z1,3,4 = x2 + y1,2 + y2,3 + y2,4 + z1,2,3 + z1,2,4 + z2,3,4 = x3 + y1,3 + y2,3 + y3,4 + z1,2,3 + z1,3,4 + z2,3,4 = x4 + y1,4 + y2,4 + y3,4 + z1,2,4 + z1,3,4 + z2,3,4 .



n−4 Furthermore, xi ∼ Bin n−4 , 1/2 , y ∼ Bin , 1/2 , and zi,j,k ∼ Bin i,j r−1 r−2 Now similarly as in Section 3 one can show that   X  n−4
4 n−4 n r−1 E(X) ∼ 2−4( r−1 ) = O(n4−3(r−1)/2 ) = o(1), 4 a a

n−4 r−3




for any r ≥ 4. Thus, the first moment method yields the statement.


, 1/2 .



5. A class of 2-weighted r-uniform hypergraphs In this section we give a sufficient condition for a hypergraph to be 2-weighted. For a hypergraph H = (V, E) and two vertices u and v in V denote by deg(u, v) the pair degree which is the number of edges in E containing {u, v}. Theorem 5.1. Let H = (V, E) be an r-uniform hypergraph of order n with maximum degree ∆ satisfying r X Y 1 p ∆· = o(1), deg(xi ) {x1 ,...,x }∈E(H) i=1 k

and for any u, v ∈ V deg(u, v) = o(deg(u)), where the asymptotic is taken in n. Then, H is 2-weighted. Proof. To each edge wep assignp 1 or 2 with probability 1/2. Since the mode of c(x) occurs with probability 2/π/ deg(x), we get r Y Pr(c(x1 ) = · · · = c(xr ) = c) = Pr(c(xi ) = c | c(x1 ) = · · · = c(xi−1 ) = c) i=1

≤

r Y

p 2/π

i=1

p deg(xi ) − o(deg(xi ))

due to our assumption on deg(u, v) = o(deg(u)). Thus, E(|{c : 0 ≤ c ≤ 2∆, ∃ {x1 , . . . , xr } such that c(x1 ) = · · · = c(xr ) = c}|) ≤

2∆ X

X

Pr(c(x1 ) = · · · = c(xr ) = c)

c=1 {x1 ,...,xk }∈E(H)

≤ 2∆

X

r Y

p 2/π p

{x1 ,...,xk }∈E(H) i=1

deg(xi ) − o(deg(xi ))

= o(1),

12

PATRICK BENNETT, ANDRZEJ DUDEK, ALAN FRIEZE, LAARS HELENIUS

by assumption completing the proof.



Corollary 5.2. Let H = (V, E) be an r-uniform hypergraph of order n and size m
k/2 with maximum degree ∆ and minimum degree δ satisfying ∆ · m · 1δ = o(1) and deg(u, v) = o(deg(u)) for any u, v ∈ V . Then, H is 2-weighted. Since trivially ∆ = O(nr−1 ) and m = O(nk ) we get for δ = Ω(n4 ) that  k/2 1 ∆·m· = O(nk−1 · nk · 1/n2k ) = o(1). δ Consequently the following holds. Corollary 5.3. Let r ≥ 5. Let H = (V, E) be an r-uniform hypergraph of order n with minimum degree Ω(n4 ) such that deg(u, v) = o(deg(u)) for any u, v ∈ V . Then, H is 2-weighted. 6. Concluding remarks In view of results of this paper we conjecture that for any r ≥ 3, any r-uniform hypergraph without isolated edges is 3-weighted. If true, then this is clearly best possible (see Figure 1). We also believe that determining whether a particular uniform hypergraph is 2-weighted is NP-complete. The latter would extend a similar result for graphs that was independently obtained by Dehghan, Sadeghi and Ahadi [3], and Dudek and Wajc [4]. It might be also of some interest to study the weightiness of H(r) (n, p) for any p = p(n). It is not difficult to see that the proof of Theorem 1.1 can be modified by replacing 1/2 by any constant p ∈ (0, 1) obtaining the same conclusion. References [1] L. Addario-Berry, K. Dalal, and B. A. Reed, Degree constrained subgraphs, Discrete Appl. Math. 156 (2008), no. 7, 1168–1174. [2] N. Alon and J. Spencer, The probabilistic method, third ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., Hoboken, NJ, 2008. [3] A. Dehghan, M.-R. Sadeghi, and A. Ahadi, Algorithmic complexity of proper labeling problems, Theoret. Comput. Sci. 495 (2013), 25–36. [4] A. Dudek and D. Wajc, On the complexity of vertex-coloring edge-weightings, Discrete Math. Theor. Comput. Sci. 13 (2011), no. 3, 45–50. [5] A. Frieze and M. Krivelevich, Packing Hamilton cycles in random and pseudo-random hypergraphs, Random Structures Algorithms 41 (2012), no. 1, 1–22. [6] R. L. Graham, M. Gr¨ otschel, and L. Lov´asz (eds.), Handbook of combinatorics. Vol. 1, 2, Elsevier Science B.V., Amsterdam; MIT Press, Cambridge, MA, 1995. [7] S. Janson, T. Luczak, and A. Ruci´ nski, Random graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, 2000. [8] M. Kalkowski, M. Karo´ nski, and F. Pfender, Vertex-coloring edge-weightings: Towards the 1-2-3conjecture, J. Comb. Theory, Ser. B 100 (2010), no. 3, 347–349. [9] , The 1-2-3 conjecture for hypergraphs, E-print: arXiv:1308.0611, 2013. [10] M. Karo´ nski, T. Luczak, and A. Thomason, Edge weights and vertex colours, J. Combin. Theory Ser. B 91 (2004), 151–157. [11] B. Seamone, The 1-2-3 conjecture and related problems: a survey, E-print: arXiv:1211.5122, 2012.

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(Andrzej Dudek, Patrick Bennett and Laars Helenius) Department of Mathematics, Western Michigan University, Kalamazoo, MI E-mail address: {patrick.bennett, andrzej.dudek, laars.c.helenius}@wmich.edu (Alan Frieze) Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA E-mail address: [email protected]