Polychromatic colorings and cover decompositions of hypergraphs

Applied Mathematics and Computation 339 (2018) 153–157

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Polychromatic colorings and cover decompositions of hypergraphs Tingting Li, Xia Zhang∗ School of Mathematics and Statistics, Shandong Normal University, Jinan 250358, PR China

a r t i c l e

i n f o

MSC: 05C15 05C65 Keywords: Property B Polychromatic coloring Cover decomposition Hypergraph

a b s t r a c t A polychromatic coloring of a hypergraph is a coloring of its vertices in such a way that every hyperedge contains at least one vertex of each color. A polychromatic m-coloring of a hypergraph H corresponds to a cover m-decomposition of its dual hypergraph H∗ . The maximum integer m that a hypergraph H admits a cover m-decomposition is exactly the longest lifetime for a wireless sensor network (WSN) corresponding to the hypergraph H. In this paper, we show that every hypergraph H has a polychromatic m-coloring if S m ≤  ln(c , where 0 < c < 1, and  ≥ 1, S ≥ 2 are the maximum degree, the minimum S2 ) size for all hyperedges in H, respectively. This result improves a result of Henning and Yeo on polychromatic colorings of hypergraphs in 2013, and its dual form improves one of Bollobás, Pritchard, Rothvoß, and Scott on cover decompositions of hypergraphs in 2013. Furthermore, we give a sufficient condition for a hypergraph H to have an “equitable” polychromatic coloring, which extends the result of Henning and Yeo in 2013 and improves in part one of Beck and Fiala in 1981 on 2-colorings (property B) of hypergraphs. © 2018 Elsevier Inc. All rights reserved.

1. Introduction In this paper, we study two interrelated problems, polychromatic colorings and cover compositions of hypergraphs. The former is a generalization of 2-colorings (Property B) of hypergraphs, and the latter is closely related to the problem of maximizing the lifetime of coverage of targets in a wireless sensor network (WSN) with battery-limited sensors. A hypergraph H = (V, E ) consists of a ground set V of vertices and a collection E of hyperedges, where each hyperedge E ∈ E is a subset of V. Throughout this paper, we consider finite hypergraphs H = (V, E ). In order to define “dual” and “shrinking” to be referred in Sections 2 and 3 (see Remark 1 and the proofs of Theorems 1.7 and 1.9), we permit E to contain multiple copies of the same subset of V and also allow hyperedges of size 0, 1 and vertices of degree 0, 1. Later, the reader will be shown that the cases for the hypergraphs containing hyperedges of size 0 or 1, or vertices of degree 0 or 1 are trivial for the two problems to be discussed. The rank of a hypergraph H is R(H ) = maxE∈E |E |, the anti-rank of H is S(H ) = minE∈E |E |. If R(H ) = S(H ) = k, that is, the size of every hyperedge in H would always be k, we say that the hypergraph H is a k-uniform hypergraph. The degree of a vertex v ∈ V (H ) is the number of hyperedges containing v in H, and is denoted by dH (v ) or simply by d (v ). A hyperedge E in H is called isolated if it does not intersect any hyperedge of H; in particular, when E = ∅, each v ∈ E has degree 1. A vertex v ∈ V is isolated, if v is not contained by any hyperedge of the hypergraph. Clearly, an isolated vertex in a hypergraph has degree 0. The maximum degree of H is denoted by (H ) = maxv∈V (H ) dH (v ), ∗

Corresponding author. E-mail addresses: [email protected], [email protected] (X. Zhang).

https://doi.org/10.1016/j.amc.2018.07.019 0 096-30 03/© 2018 Elsevier Inc. All rights reserved.

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the minmum degree of H is denoted by δ (H ) = minv∈V (H ) dH (v ). A hypergraph in which all vertices have the same degree k is called a k-regular hypergraph. Throughout this paper, we denote the class of k-regular k-uniform hypergraphs by Hk . A hypergraph H is 2-colorable (or has Property B) if it has a 2-coloring of the vertices with no monochromatic hyperedges. Alon and Bregman [1] established the following result. Theorem 1.1 [1]. Every hypergraph in Hk is 2-colorable, provided that k ≥ 8. Furthermore, Vishwanathan [8] obtained the following result. Henning and Yeo [7] provided a short proof. Theorem 1.2 [8]. Every hypergraph in Hk is 2-colorable, provided that k ≥ 4. Noting that Fano plane is in H3 and but not 2-colorable, the bound above for k is sharp. Chen et al. [5] discussed a class of 2-colorings of hypergraphs in which each color appears at least two times on each hyperedge. Theorem 1.3 [5]. Let H be a hypergraph in which every hyperedge contains at least k (k ≥ 4) vertices and meets at most d other hyperedges. If e(k + 1 )2−k (d + 2 ) ≤ 1, then H has a 2-coloring such that each hyperedge contains at least two vertices of each color. The below result, due to Henning and Yeo [7], shows us some bounds for k when we want each color to appear at least r + 1 times on each hyperedge of a hypergraph in Hk . Theorem 1.4 [7]. Let k ≥ 2, r ≥ 2 be two integers and H ∈ Hk . Then H has a 2-coloring such that each hyperedge contains at least r + 1 vertices of each color if one of the following conditions holds.

 √ k ln(k 2e). (b) k ≥ 2r + 3 r ln(r ) + 44.03.  (c) k ≥ 2r + 4 r ln(r ) + 14.04. (a) r ≤ k/2 −

Here conditions (a )–(c ) imply that k ≥ 24, k ≥ 52, k ≥ 23, respectively. A result of Beck and Fiala [2] is related to the problem to partition the vertices of a hypergraph into two parts in such a way that each part is relatively “equitable” on each hyperedge. Theorem 1.5 [2]. Let H be a hypergraph with maximum degree , where  ≥ 2. Then, H has a 2-coloring such that each hyperedge E ∈ E contains at least |E |/2 −  + 1 vertices of each color. Let m be a positive integer. An m-coloring of the vertices of a hypergraph is a polychromaticm-coloring if every hyperedge contains at least one vertex of each color. This is clearly a generalization of 2-colorings of a hypergraph. A natural question is how large m could be if we want a hypergraph to have a polychromatic m-coloring. Henning and Yeo [7] gave the following result. Theorem 1.6 [7]. Let k ≥ 2, m ≥ 2 be two integers. If m ≤

k , ln(k3 )

then every hypergraph H ∈ Hk has a polychromatic m-coloring.

Obviously, if a hypergraph H has a polychromatic m-coloring, then H has a polychromatic l-coloring for each 1 ≤ l ≤ m. Therefore, it is interesting to find the maximum m that the hypergraph H admits a polychromatic m-coloring, which is called the polychromatic number of H and denoted by p(H). So Theorem 1.6 means that p(H ) ≥  ln(kk3 )  for each H ∈ Hk (k ≥ 2). Trivially, p(H ) = 1 for each hypergraph H with anti-rank 1. Next, we concentrate on the hypergraphs with anti-rank at least 2. One of our main results is below.

Theorem 1.7. Let S ≥ 2 and  ≥ 1 be two integers, and let H be a hypergraph with maximum degree at most  and anti-rank S at least S. Then p(H ) ≥  ln(c , where c = ln(ee+1 . S2 ) S2 ) Note that 0 < c < 1.5582 < e here. Immediately, we have the following result for hypergraphs H in Hk . k Corollary 1.8. Let H be a hypergraph in Hk and k ≥ 2. Then p(H ) ≥  ln(ck 3 ) , where c =

When k ≥ 15, we have

k ln(ck3 )

e+1 . 1+ln(k3 )

≥ 2 and 0 < c < 0.41. Note that the conditions in Theorem 1.6 implies that k ≥ 17. Thus,

Corollary 1.8 improves Theorem 1.6. Furthermore, we obtain a result on “equitable” polychromatic colorings of hypergraphs. Theorem 1.9. Let S ≥ 2 and  ≥ 1 be two integers, and let H be a hypergraph with maximum degree at most  and anti-rank S at least S. Then, for every positive integers m ≤ ln(e and r = ln(eS2 ), H has a polychromatic m-coloring such that every S2 ) hyperedge in H contains at least r vertices of each color. For each fixed , ditions: (a) m = 2 =

S ≥ ln(eS2 ) S  ln(e  ; S2 )

2 when S is large enough. So this result improves Theorem 1.5 under the following con(b) the gap between the rank and the anti-rank of a hypergraph is less than 2 − 4. In

T. Li, X. Zhang / Applied Mathematics and Computation 339 (2018) 153–157

particular, when k ≥ 20,

k ln(ek3 )

155

≥ 2, which means that we could get a more “equitable” m-coloring for each hypergraph in

Hk . Thus, Theorem 1.9 extends Theorem 1.6. In Section 2, we will show the relation between polychromatic colorings and cover compositions of hypergraphs. In Section 3, we give the proofs, mainly by way of Lovász Local Lemma. 2. Cover decompositions of hypergraphs A subfamily Ei of E in a hypergraph H = (V, E ) is called a cover in H if ∪E∈Ei E = V . A cover m-decomposition of a hypergraph H is a partition of E into m covers in H, i.e. E = m E and ∪E∈Ei E = V . The maximum integer m that the hypergraph i=1 i H admits a cover m-decomposition is called the cover-decomposition number of H and denoted by cd(H). A hypergraph H can model a collection of sensors, with each hyperedge E ∈ E corresponding to a sensor which can monitor the vertices in E ⊆ V. Since monitoring all vertices (targets) of V takes a cover in H, cd(H) is the maximum possible “coverage” of V if each sensor can only be turned on for a single time unit. Hence cd(H) is exactly the longest lifetime for a WSN corresponding to the hypergraph H [4]. Let R, δ be two nonnegative integers and let Hyp(R, δ ) denote the family of hypergraphs with rank at most R and minimum degree at least δ , and

cd (R, δ ) = min{cd (H )| H ∈ Hyp(R, δ )}. Bollobás et al. [3] gave lower and upper bounds on cd (R, δ ). Theorem 2.1 [3]. (i) For all R, δ , we have cd (R, δ ) ≥ δ /(ln R + O(ln ln R )). (ii) For all R ≥ 2, δ ≥ 1, we have cd (R, δ ) ≤ max{1, O(δ / ln R )}. (iii) For each sequence R, δ → ∞ with δ = ω (ln R ), we have cd (R, δ ) ≤ (1 + o(1 ))δ / ln(R ). In particular, they obtained the following result. Theorem 2.2. [3] For all R, δ , cd (R, δ ) ≥ δ / ln(eRδ 2 ). The dual of a hypergraph H = (V, E ) with V = {x1 , x2 , . . . , xn } and E = (E1 , E2 , . . . , Em ) is a hypergraph H∗ whose vertices e1 , e2 , . . . , em correspond to the hyperedges of H, and whose hyperedges Xi = {e j | xi ∈ E j in H}, i = 1, 2, . . . , n. Clearly, (H ∗ )∗ = H and (H ∗ ) = R(H ), δ (H ∗ ) = S(H ), R(H ∗ ) = (H ), S(H ∗ ) = δ (H ). So a polychromatic m-coloring of H∗ corresponds to a cover m-decomposition of H, and cd (H ) = p(H ∗ ). Remark 1. A hypergraph may have a polychromatic m-coloring even if it contains some isolated vertices. In Theorems 1.7 and 1.9, we permit hypergraphs to contain vertices with degree 0 or 1. By duality, we also permit hypergraphs to contain hyperedges with size 0 or 1. An isolated hyperedge E with |E| ≥ 2 in a hypergraph H corresponds to a vertex e which is contained in |E| multiple hyperedges with size 1 in H∗ . For integer t ≥ 2, t isolated vertices in a hypergraph H correspond to t multiple hyperedges with size 0 in H∗ . In order to define “dual” and “shrinking” (shrinking a hyperedge e in a hypergraph means to replace it with some e ⊂ e) to be used later, we permit E to contain multiple copies of the same subset of V and also allow hyperedges of size 0 or 1 and vertices of degree 0 or 1 in this paper. On the other hand, it is trivial to verify that cd (R, δ ) = 0 when δ = 0 and cd (R, δ ) = 1 when δ = 1, by duality, p(, S ) = 0 when S = 0 and p(, S ) = 1 when S = 1. So we can concentrate on the hypergraphs with S ≥ 2 or δ ≥ 2 in the rest of the paper. By duality, Theorems 1.7 and 1.9 could be rewritten as below. δ Theorem 1.7 (stated in the dual) For all integers δ ≥ 2 and R ≥ 1, cd (R, δ ) ≥  ln(cR , where c = δ2 )

e+1 . ln(eRδ 2 )

Theorem 1.9 (stated in the dual) Let δ ≥ 2 and R ≥ 1 be two integers and let H be a hypergraph with rank at most R and δ minimum degree at least δ . Then, for every positive integers m ≤ ln(eR and r = ln(eRδ 2 ), H has a cover m-decomposition δ2 ) E = m E such that δ (V, Ei ) ≥ r for each 1 ≤ i ≤ m. i=1 i

Note that 0 < c < 1.5582 < e when δ ≥ 2 and R ≥ 1. So Theorem 1.7 improves Theorem 2.2. 3. Proofs of Theorems 1.7 and 1.9 In this section, we give the proofs of our main results by using the useful operation shrinking and the Lovász Local Lemma. Theorem 3.1 Lova´ sz Local Lemma [6]. Let A = {A1 , A2 , . . . , An } be a set of (typically bad) events in a probability space. Suppose that each event Ai is mutually independent of a set of all but at most d of the other events, and Pr (Ai ) ≤ p for all 1 ≤ i ≤ n. If

ep(d + 1 ) < 1 then Pr (

n

i=1

Ai ) > 0.

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T. Li, X. Zhang / Applied Mathematics and Computation 339 (2018) 153–157

Theorem 1.7. Let S ≥ 2 and  ≥ 1 be two integers, and let H be a hypergraph with maximum degree at most  and anti-rank at S least S. Then p(H ) ≥  ln(c , where c = ln(ee+1 . S2 ) S2 ) S Proof. Let m =  ln(c . Since S ≥ 2, we have p(H) ≥ 1. So we just need discuss the cases with m ≥ 2. Next, we will show S2 )

how to determine c such that H has a polychromatic m-coloring and m as large as possible. Let E (H ) = {e1 , e2 , . . . , e|E (H )| }. If H is not S-uniform, we can shrink some ej to ej until |ej | = S when |ej | > S. It is easy to see that undoing shrinking preserves the property of being a hyperedge containing m colors. This operation is useful because it bounds the dependence degree later. So, next we assume that H is S-uniform. Consider a random coloring of H, in which every vertex is independently and uniformly assigned a color from {1, 2, . . . , m}. We define the bad event Aj to be the event that one color is missing from ej . Thus,



Pr (A j ) ≤ m · 1 −
1

1 m

S/m

e



S



Since 1 −

1 x

x

<

1 e



for all x > 1

=p

Note that Aj and Aa are independent unless ea intersects ej . As each hyperedge intersects at most S( − 1 ) other hyperedges, Aj depends on at most S( − 1 ) other events. That is, d = S( − 1 ). By the Lova´ sz Local Lemma, we show that, with a positive probability, no bad events happens if the following holds.



ep(d + 1 ) = e ·

m·

 1 S/m  e

· [S( − 1 ) + 1] < 1.

(1)

Let c = te, where 0 < t < 1 will be fixed later. Thus,



e·

m·

 1 S/m  e

· [S( − 1 ) + 1] < e ·

S · ln(teS2 )

 1 ln(teS2 ) e

· S

1 eS · · S ln(teS2 ) teS2 1 = t ln(teS2 ) 1 = t ln(t ) + t ln(eS2 ) =

(2)

If (2) is at most 1, then (1) holds. For each

t≥

e+1 , e ln(eS2 )

there is

t ln(eS2 ) ≥ 1 +

1 . e

Also, since t ln(t ) ≥ − 1e for all t > 0,

t ln(t ) + t ln(eS2 ) ≥ −

1 + t ln(eS2 ) ≥ 1. e

Since S ≥ 2 and  ≥ 1, it is easy to verify that 0 <

e+1 e ln(eS2 )

< 0.5733. Clearly, t =

e+1 e ln(eS2 )

So (1) holds. S Therefore, H has a polychromatic m-coloring when 2 ≤ m =  ln(c , where c = S2 )

makes

e+1 . ln(eS2 )

S ln(cS2 )

as large as possible.



Theorem 1.9. Let S ≥ 2 and  ≥ 1 be two integers, and let H be a hypergraph with maximum degree at most  and anti-rank S at least S. Then, for every positive integers m ≤ ln(e and r = ln(eS2 ), H has a polychromatic m-coloring such that every S2 ) hyperedge in H contains at least r vertices of each color. Proof. The case for m = 1 is trivial. So assume that 2 ≤ m ≤

S . ln(eS2 )

Let E (H ) = {e1 , e2 , . . . , e|E (H )| }. As described in the proof of Theorem 1.7, we can assume that H is S-uniform (after some necessary hyperedge-shrinking). Consider a random coloring of H, in which every vertex is independently and uniformly assigned a color from {1, 2, . . . , m}. Define the bad event Aj to be the event that the hyperedge ej contains at most (r − 1 ) vertices of a color. Then

Pr (A j ) ≤ m ·



1−

1 m

S



+ 1−

1 m

S−1 1 ·

m



+ ··· + 1 −

1 m

S−r+1  1 r−1  ·

m

T. Li, X. Zhang / Applied Mathematics and Computation 339 (2018) 153–157

≤m·



1−



1 m

=m·r· 1−
S



+ 1−

 1 S



e

 1 ln(eS2 )

S

Since

m

 1 S/m

1 m





+ ··· + 1 −



1−

 1 x x

Since 2 ≤ m ≤

<

157

 1 S m 1 e

 for all x > 1

S ln(eS2 )

, r = ln(eS2 )



e

1 e S

= p < 1.

Note that Aj and Aa are independent unless ea intersects ej . As each hyperedge intersects at most S( − 1 ) other hyperedges, Aj depends on at most S( − 1 ) other events. That is, d = S( − 1 ). Thus,

ep(d + 1 ) = e ·

1 1 · [S( − 1 ) + 1] < e · · (S) = 1. e S e S

By the Lovász Local Lemma, with a positive probability, no bad events happens.



Acknowledgments The authors would like to thank the referees for their careful reading and valuable comments. Also, we thank Professor Gexin Yu for his helpful suggestions. This work is supported by the National Natural Science Foundation of China (Nos. 11701342 and 11626148) and the Natural Science Foundation of Shandong Province, China (Nos. ZR2014JL001 and ZR2016AQ01). References [1] [2] [3] [4] [5] [6] [7] [8]

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