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Brief Research Report ARTICLE

Front. Phys., 16 June 2020 | https://doi.org/10.3389/fphy.2020.00197

Generalization of the Cover Pebbling Number for Networks

Zheng-Jiang Xia* and Zhen-Mu Hong
  • School of Finance, Anhui University of Finance & Economics, Bengbu, China

Pebbling can be viewed as a model of resource transportation for networks. We use a graph to denote the network. A pebbling move on a graph consists of the removal of two pebbles from a vertex and the placement of one pebble on an adjacent vertex. The t-pebbling number of a graph G is the minimum number of pebbles so that we can move t pebbles on each vertex of G regardless of the original distribution of pebbles. Let ω be a positive function on V(G); the ω-cover pebbling number of a graph G is the minimum number of pebbles so that we can reach a distribution with at least ω(v) pebbles on v for all vV(G). In this paper, we give the ω-cover pebbling number of trees for nonnegative function ω, which generalized the t-pebbling number and the traditional weighted cover pebbling number of trees.

Mathematics Subject Classification: 05C99, 05C72, 05C85.

1. Introduction

Pebbling in graphs was introduced by Chung [1]. It can also be viewed as a model of resource transportation for networks. Let G be a simple connected graph; we use V(G) and E(G) to denote the vertex set and edge set of G, respectively. d(u, v) is the distance of u and v, and we write u~v if they are adjacent. N(v) = {u|u ~ v} is the neighbor of v, and d(v) = |N(v)| is the degree of v. Let H be a subgraph of G; we use dH(v) to denote the degree of v in H.

A pebble distribution D on G is a function D:V(G) → N (N is the set of nonnegative integers), where D(v) is the number of pebbles on v, |D|=vV(D)D(v) is the size of D.

A pebbling move consists of the removal of two pebbles from a vertex and the placement of one pebble on an adjacent vertex. Let D and D′ be two pebble distributions of G. If so, we say that D contains D′ if D(v) ≥ D′(v) for all vV(G), and D′ is reachable from D if there is some sequence (probably empty) of pebbling moves (a pebbling sequence in short) starting from D and resulting in a distribution, which contains D′. For a graph G and a vertex v, we call v a root (or target vertex) if the goal is to place pebbles on v. If t pebbles can be moved to v from D by a sequence of pebbling moves, then we say that D is t-fold v-solvable, and v is t-reachable from D. If D is t-fold v-solvable for every vertex v, we say that D is t-solvable.

The t-pebbling number of a graph G, denoted by ft(G), is the smallest number m such that every distribution with size m is t-solvable. While t = 1, we use f(G) instead of f1(G), which is called the pebbling number of G.

For any two graphs G and H, we define the Cartesian product G × H to be the graph with the vertex set V(G × H) and edge set the union of {((a, v), (b, v))|(a, b) ∈ E(G), vE(H)} and {((u, x), (u, y))|uV(G), and(x, y) ∈ E(H)}.

To determine the pebbling number of a general graph G is difficult. The problem of whether a distribution is v-solvable for some vV(G) was shown to be NP-complete [2, 3]. The problem of deciding whether the pebbling number of a graph G is less than k was shown to be Π2P-complete [3]. The pebbling numbers of trees [4], cycles [5], hypercubes [1], and so on are determined. A conjecture called Graham's Conjecture is given by Chung [1].

Conjecture 1.1. (Graham's Conjecture) Let G and H be two connected graphs; the pebbling number of the Cartesian product of G and H satisfies:

f(G×H)f(G)f(H).

There are many results about Graham's Conjecture [610], while this conjecture is still open.

Definition 1.2. Let ω be a nonnegative function on V(G) and D a distribution on V(G). We say D is ω-solvable (or D solves ω) if we can reach a distribution D* from D, by a sequence of pebbling moves, so that D*(v) ≥ ω(v) for all vV(G). The ω-cover pebbling number of G, denoted by γω(G), is the minimum number m so that every distribution D with size m is ω-solvable.

Definition 1.3. Let ω be a positive function on V(G); define

sω(v)=uV(G)ω(u)2d(u,v),

and

sω(G)=maxvV(G)sω(v).

The ω-cover pebbling number of a graph G has been determined for positive ω by [11].

Theorem 1.4. ([11]) Let ω be a positive weight function on V(G); the ω-cover pebbling number of G is

γω(G)=sω(G).

From Theorem 1.4, we can get

Theorem 1.5. ([11]) Let ω1 be a positive function on G and ω2 be a positive function on H. The function ω on G × H is given by ω((g, h)) = ω1(g2(h), where gV(G) and hV(H), then γω(G × H) = γω1(Gω2(H).

We first generalize the definition of sω(T) while ω is a nonnegative function on a tree T. We will give the definition of path partition in the next section.

Definition 1.6. Given a tree T and a nonnegative function ω for each vertex vV(T), and let Tω(v) be the minimum subtree of T containing v and W: = {u:ω(u) > 0}. We give each edge in T\E(Tω(v)) a direction toward Tω(v) to get a directed graph, which is denoted by T\E(Tω(v)), and (a1, …, an) is the size of the maximum path partition of T\E(Tω(v)). We define

sω(v)=uWω(u)2d(u,v)+i=1n2ai-n.

and

sω(T)=maxvV(T)sω(v).

Note that while ω is positive, then the two definitions of sω(T) are the same. Definition 1.6 is thus a generalization of Definition 1.3. In this paper, we generalize Theorem 1.4 while T is a tree and ω is nonnegative. Thus, our main result is as follows

Theorem 1.7. Let T be a tree with a nonnegative weight function ω on V(T). The ω-cover pebbling number of T is

γω(T)=sω(T).

Theorem 1.8. Let T be a tree with a nonnegative weight function ω on V(T). If |W| = 1, then Theorem 1.7 is equivalent to Theorem 2.2.

Proof. If |W| = 1, assume that ω(v) = t, and ω(u) = 0 for uv. We will show that ft(T, v) = sω(T).

Assume the size of a maximum path partition of Tv is (a0, a1, …, an), and d(v, v0) = a0, P0 is the longest directed path from v0 to v. Then (a1, …, an) must be the size of a maximum path partition in Tv\P0. So ft(T, v) = sω(v0) ≤ sω(T).

Assume sω(T) = sω(v1), and d(v1, v) = a0. Let P0 be the path connected v1 and v, then Tω(v1) = P0; assume (a1, …, an) is the size of the maximum path partition of T\E(Tω(v)) = T\E(P0), so α = (a0, a1, …, an) is a path partition of Tv, and sα = sω(v1) by Corollary 2.3 and ft(T, v) ≥ sω(v1) = sω(T).

Definition 1.9. ([12]) Given a sequence S of pebbling moves on G, the transition digraph obtained from S is a directed multigraph denoted T(G, S) that has V(G) as its vertex set. Each move sS along edge uv (move off two pebbles from u and add one on v) is represented by a directed edge uv.

The following lemma is useful in the following sections.

Lemma 1.10. ([12], No-Cycle Lemma) Let S be a sequence of pebbling moves on G, reaching a distribution D. Then there exists a sequence S* of pebbling moves, thus reaching a distribution D* where

1. On each vertex v, D*(v) ≥ D(v);

2. T(G, S*) does not contain any directed cycles.

2. Preliminaries

We first introduce the path partition and the pebbling number of trees.

Definition 2.1. ([4]) Given a root v of a tree T, then we can view T as a directed graph Tv with edges directed toward v. A path partition is a set of nonoverlapping directed paths in which the union is Tv. A path partition is said to majorize another if the non-increasing sequence of the path size majorizes that of the other (that is (a1, a2, …, ar) > (b1, b2, …, bt) if and only if ai > bi, where i = min{j:ajbj}). A path partition of a tree Tv is said to be maximum if it majorizes all other path partitions. Note that, in this paper, the sequence of the size of a path partition is always non-increasing.

Note: By the definition of the maximum path partition, we can give a way to determine the size of the maximum path partition. First, we choose the longest directed path P1 in Tv, with length a1. Then, we choose the longest directed path P2 in Tv\E(P1), with length a2, and so on. Moreover, it should be noted that the maximum path partition may not be unique, but the size of the maximum path partition must be unique.

Moews [4] found the t-pebbling number of trees by a path partition.

Theorem 2.2. ([4]) Let T be a tree, vV(T), and (a1, …, an) be the size of the maximum path partition of Tv. Then,

ft(T,v)=t2a1+i=2n2ai-n+1,
ft(T)=maxvV(T)ft(T,v).

Corollary 2.3. Let T be a tree, vV(T), and α = (a1, …, an) be the size of a path partition of Tv, sα:=t2a1+i=2n2ai-n+1. Then,

ft(T,v)=maxαsα.

Proof. Let α0 be the size of the maximum path partition of Tv. Then, ft(T,v)=sα0maxαsα.

Assume P1, P2, …, Pn is a path partition of Tv, and the length of Pi is ai for 1 ≤ in. Note that for each Pi we should assume the two endpoints vi and vi satisfy d(vi,v)>d(vi,v). We put t2a1-1 pebbles on v1 and 2ai-1 pebbles on vi for 2 ≤ in; it is clear that t pebbles cannot be moved to v from this distribution. Thus, for each α, sα − 1 < ft(T, v), so sαft(T, v) so maxαsαft(T,v).

Definition 2.4. Let C be a generalized distribution on G, where C(v) is an integer (may be negative) for all vV(G). A pebbling move on G consists of the removal of two pebbles from a vertex v (with C(v) ≥ 2) and the placement of one pebble on an adjacent vertex.

In the following, a distribution D means that D(v) ≥ 0, and a generalized distribution C means C(v) is an integer for all vV(G).

Definition 2.5. A pebbling move from u to v under a distribution D is executable if D(u) ≥ 2. A pebbling sequence S is a finite set of pebbling moves, assuming S = (S1, …, Sk), where Si is a pebbling move for 1 ≤ ik, and the pebbling move Si is in front of Sj if 1 ≤ i < jk. We say the pebbling sequence S executable, if Si is executable for 1 ≤ ik.

Definition 2.6. Let ω be a nonnegative function on V(G) and C be a generalized distribution on V(G). We say C is ω-solvable, if we can reach a distribution C* from C, by a sequence of pebbling moves so that C*(v) ≥ ω(v). In particular, if ω(v) = 0 for all vV(G), then we say that C is 0-solvable.

Lemma 2.7. Let D be a distribution on a graph G and ω be a nonnegative function on V(G), C: = D − ω. Then, D is ω-solvable if and only if C is 0-solvable.

Proof. If C is 0-solvable, let δ be an executable pebbling sequence that reaches a distribution D* so that D* > 0 from C. It is then clear that δ is also an executable pebbling sequence that can reach a distribution D′ so that D′ = D* + ω > ω from D. Thus D is ω-solvable.

On the other hand, if D is ω-solvable, by Lemma 1.10, there exists a pebbling sequence S reaching a distribution D* with D*(v) ≥ ω(v), and T(G, S) does not contain any direct cycle. We can thus give a sequence of the vertices of G, as (v1, v2, …, vn), so that each directed edge vivj in T(G, S) satisfies i < j. We can thus rearrange the sequence of pebbling moves S along the order (v1, v2, …, vn). It means we first choose all pebbling moves in S that remove pebbles from v1, choose all pebbling moves in S that remove pebbles from v2, and so on, and we denote this sequence of pebbling moves by S′. We will show that S′ is an executable pebbling sequence that reach D* − ω from C.

In S′, for each vertex vV(G), the pebbling moves that move pebbles to v are in front of the pebbling moves that remove pebbles from v. We may thus assume that, for each vertex vi, we first move αi pebbles from other vertices to vi and then remove βi pebbles from vi.

We only need to show that, for each viV(G), the sequence of pebbling moves that removes βi pebbles from vi in S′, denoted by σi, is executable. We use induction on i. If i = 1, and we can then get D(v1)-β1=D*(v1)ω(v1)D(v1)-ω(v1)β1C(v1)β1, and so σ1 is executable.

Assume σh is executable for h < i. By induction, the pebbling sequence that moves αi pebbles to vi is executable. Moreover, we can get D(vi)+αi-βi=D*(vi) D(vi)+αi-ω(vi)-βi= D*(vi)-ω(vi)0D(vi)-ω(vi)+ αiβiC(vi)+αiβi. Thus σi is executable.

So S′ is an executable pebbling sequence that reaching D* − ω from C. Note that D* − ω ≥ 0, and this completes the proof.

Definition 2.8. Let D be a distribution on a tree T and ω be a nonnegative function on V(T). C: = D − ω is called the induced generalized distribution from D and ω of T. Let v be a leaf of T and u be its neighbor in T. The induced generalized distribution C′ on T\v is given: if C(v) ≥ 0, then C′(u) = C(u) + C(v)/2, and if C(v) < 0, then C′(u) = C(u) + 2C(v), keeping C′(x) = C(x) unchanged for all xu.

Lemma 2.9. Let D be a distribution on a tree T and ω be a nonnegative function on V(T). C: = D − ω, v is a leaf of T, and C′ is the induced generalized distribution from D and ω of T\v. Then, C is 0-solvable in T if and only if C′ is 0-solvable in T\v.

Proof. Firstly, we assume C is 0-solvable in T, and there is a pebbling sequence σ reaching a distribution C* from C with C*(x) ≥ 0 for each xV(T).

Case 1.1. C(v) ≥ 0. By Lemma 1.10, we may assume that no pebble has been moved from u to v; at most, therefore, C(v)/2 pebbles can be moved from v to u. We may assume the first step of σ is to move C(v)/2 pebbles from v to u, so the left steps makes C′ solve 0 on T\v.

Case 1.2. C(v) < 0. By Lemma 1.10, we may assume that no pebble has been moved from v to u. So we may assume the last step of σ is to move −C(v) pebbles from u to v, and so the steps before it makes C′ solve 0 on T\v.

Secondly, we assume C′ is 0-solvable in T\v, and there is a pebbling sequence δ reaching a distribution C* from C′ with C*(x) ≥ 0 for each xV(T\v).

Case 2.1. C(v) ≥ 0. First, we move C(v)/2 pebbles from v to u, and the left steps are just δ; this sequence makes C solve 0.

Case 2.2. C(v) < 0. After the pebbling sequence δ, we move −C(v) pebbles from u to v; this sequence makes C solve 0.

Notations: Assume T* is a subtree of T, then T* can be obtained from T by deleting the leaves of the subtree of T (the vertex with degree one) one by one. For each subtree T* of T, therefore, we can get an induced generalized distribution C*. In particular, for each vertex vV(T), let Tv be a subtree containing v and all of its neighbors. We use Cv to denote the induced generalized distribution from D and ω of Tv and C^(v) to denote the induced generalized distribution of {v}.

Corollary 2.10. Let D be a distribution on a tree T, ω be a nonnegative function on V(T), and C^(v) be the induced generalized distribution from D and ω of {v}. D is not ω-solvable is equivalent to C^(v)<0 for each vV(T).

Proof. From Lemma 2.7 and Lemma 2.9, the result follows by induction.

Lemma 2.11. Let D be a distribution on a tree T, which is not ω-solvable with |D| = γω(T) − 1. For each vertex xV(T), which is not a leaf of T, there exists a vertex yN(x), so that Cx(y) ≥ 0.

Proof. If Cx(x)<0, for all x′ ∈ N(x), assume y, zN(x) with Cx(z) ≤ Cx(y) < 0. We delete all other vertices to the left T1 = yxz and its induced generalized distribution C1. Then, C1(y) = Cx(y), C1(z) = Cx(z) and C^(x)=C1(x)+2C1(y)+2C1(z)-1 by Corollary 2.10. Note that C1(x)=D(x)-w(x)+xN(x),x{y,z}2Cx(x). Thus, C1(x) − D(x) ≤ 0 and C1(x) + 2C1(z) − D(x) ≤ 0. Now, we remove D(x) pebbles from x and put D(x) + 1 pebbles on y to get a new distribution D′ with |D′| = |D| + 1. The induced generalized distribution from D′ and ω of {y} is denoted by C^(y). Then, C^(y)=(C1(y)+D(x)+1)+ 2(C1(x)+2C1(z)-D(x))= (C1(x)+2C1(y)+2C1(z))+(C1(z)-C1(y)) +C1(z)+ (C1(x)-D(x))+1-1+0-1+0+1=-1, and so D′ is not ω-solvable by Corollary 2.10, a contradiction to |D|=γω(T), and we are done.

Theorem 2.12. Let ω be a nonnegative function on V(T) and D be a distribution that is not ω-solvable with |D| = γω(T) − 1. All pebbles are then distributed on the leaves of T.

Proof. If D(x) > 0 for some vertex xV(T), which is not a leaf, then N(x) has at least two vertices. By Lemma 2.11, there exists a vertex yN(x) with Cx(y) ≥ 0. We first show that there exists a vertex zN(x) with Cx(z) < 0.

If not, that means for all vN(x), Cx(v) ≥ 0. Note that D(x) > 0, and thus C^(x)=D(x)+vN(x)Cx(v)/2>0. By Corollary 2.10, D is ω-solvable, a contradiction. Thus, there exists a vertex zN(x) with Cx(z) < 0.

Let T1 = yxz be the subtree of T, with induced generalized distribution C1. Then, C1(z) = Cx(z), C1(y) = Cx(y), and C^(x)=C1(x)+C1(y)/2+2C1(z)<0.

Now, consider the new distribution D*, with D*(y) = D(y) + D(x)+1, D*(x) = 0, and D*(v) = D(v); |D*|=γω(T). The induced generalized distribution from D* and ω of {x} is given by C*^(x)=(C1(x)-D(x))+(C1(y)+D(x)+1)/2+2C1(z).

If D(x) = 1, then C*^(x)=C1(x)+C1(y)/2+2C1(z)=C^(x)<0;

If D(x) ≥ 2, then C*^(x)C1(x)-D(x)+C1(y)/2+D(x)/2+1 +2C1(z)=C^(x)+1-D(x)/2C^(x)<0.

By Corollary 2.10, D* is not ω-solvable, a contradiction to |D*|=γω(T). This completes the proof.

From Theorem 2.12, for a given integer p with p < γω(T), there must exist a distribution D, which is not ω-solvable with |D| = p, and all pebbles are distributed on the leaves of T.

3. The Generalization of the Cover Pebbling Number on Trees

Assume that sω(v0) = sω(T) for some v0V(T); it should be noted that T\E(Tω(v0)) is a directed graph. We define dω(u, l) to be the length of the maximal path containing u in all maximum path partitions of T\E(Tω(v0)). If ω is clear, then we use d(u, l) for short (note that d(u, l) maybe 0). Let Pα be a maximal path partition of T\E(Tω(v0)); then, dω(u,l)=maxPα{|P|:uP,PPα}.

Lemma 3.1. Assume that sω(v0) = sω(T) for some v0V(T); then for each vertex uV(T) and d(u, v0) ≥ d(u, l).

Proof. Assume u, vV(T). There is exactly one subpath of T with endpoints u and v, and we denote this path by Puv. We thus have Puv = Pvu.

If |W| = 1, we may assume that ω(v) = t, and ω(u) = 0 for uv. By the proof of Theorem 1.8, we know that ft(T, v) = sω(v0). Let (a1, a2, …, an) be the size of the maximum path partition of Tv. Then d(v,v0)=maxuV(T)d(v,u)=a1. Assume P1 is the maximal path containing u in Tv\Pv0,v, and P1Pv0v = v1. The length of Pv0v (P1) is thus a1 (d(u, l)) and d(v1, v0) ≤ d(u, v0). If d(u, v0) < d(u, l), then d(v1, v0) < d(u, l), and we get a path P1Pv1v with length a1d(v1, v0) + d(u, l) > a1, a contradiction to the maximum of a1, and thus d(u, v0) ≥ d(u, l).

If |W| ≥ 2, we only need to show it while uV(Tω(v0)).

If d(u, v0) < d(u, l) for some uV(Tω(v0)), there exists a leaf v1 in T\E(Tω(v0)) so that d(u, l) = d(u, v1), and we will show that sω(v1) > sω(v0).

Let TC(v) be the component of T\u containing the vertex v. We thus have TC(v1) ∩ W = ∅.

Case 1. TC(v0) ∩ W ≠ ∅.

Assume w1TC(v0) ∩ W, then d(w1, v1) ≥ d(u, v1) + 1 and

d(w1,v1)-d(w1,v0)d(u,v1)-d(u,v0)+23.

Note that T\E(Tω(v0)Pv1u)T\E(Tω(v1)). So

sω(v1)-sω(v0)xWω(x)(2d(x,v1)-2d(x,v0))-2d(u,v1)ω(w1)(2d(w1,v1)-2d(w1,v0))-2d(u,v1)2d(w1,v1)-2d(w1,v0)-2d(u,v1)2d(w1,v1)-2d(w1,v1)8-2d(w1,v1)2=3·2d(w1,v1)8>0.

Hence, sω(v1) > sω(v0), which is a contradiction to sω(v0) = sω(T).

Case 2. TC(v0) ∩ W = ∅.

Let τω(v)=xWω(x)2d(x,v). If so, then τω(v0)=2d(u,v0)τω(u), and τω(v1)=2d(u,v1)τω(u). For |W| ≥ 2, τω(u)20+21=3.

Note that T\E(Tω(v0)Pv1u)T\E(Tω(v1)). So

sω(v1)-sω(v0)2d(u,v1)τω(u)-2d(u,v0)τω(u)-2d(u,v1)=τω(u)(2d(u,v1)-2d(u,v0))-2d(u,v1)3(2d(u,v1)-2d(u,v0))-2d(u,v1)3(2d(u,v1)-2d(u,v1)2)-2d(u,v1)=2d(u,v1)2>0.

Hence, sω(v1) > sω(v0), which is a contradiction to sω(v0) = sω(T), and this completes the proof.

Corollary 3.2. Let ω be a nonnegative function in V(T), for some vW, and ω′ be a nonnegative function satisfying ω′(v) = ω(v)−1, ω′(u) = ω(u) for other vertices in T. If so, then

sω(T)sω(T)+2dω(v,l).

Proof. Assume that there exist v1 and v2, so that sω(v1) = sω(T) and sω(v2)=sω(T).

By the definition of sω(v), if ω(v) ≥ 2, then dω(v,l)=dω(v,l), we have

sω(T)=sω(v1)sω(v2)          =sω(v2)+2d(v,v2)           sω(v2)+2dω(v,l)  (by Lemma 3.1)          =sω(T)+2dω(v,l).

If ω(v) = 1, the difference between T\Tω(v1) and T\Tω(v2) is just the length of the maximal path containing v, we have

sω(T)=sω(v1)sω(v2)                           =sω(v2)+2d(v,v2)+2dω(v,l)-2dω(v,l)                           sω(v2)+2dω(v,l)  (by Lemma 3.1)                           =sω(T)+2dω(v,l).

The proof of Theorem 1.7:

The lower bound holds clearly, as we put 2ai-1 pebbles on the leaf of each path for 1 ≤ in (no pebble can then be moved to Tω(v)), and uSw(u)2d(u,v)-1 pebbles on v, obviously it is not ω-solvable.

For the upper bound, it holds if |ω| = 1 or |W| = 1 by the proof of Theorem 1.8. It also holds for |T| ≤ 2 by Theorem 2.2 and Theorem 1.4. We may thus assume that |ω| ≥ 2, |W| ≥ 2, and |T| ≥ 3.

If the result is false for some T and ω, then we choose one counterexample T and its weight ω so that |T| and |ω| are both minimal. It means the upper bound holds for T′ and its weight ω′ if |T′| < |T| or |ω′| < |ω|.

Let D be a distribution on T, which is not ω-solvable with size sω(T). By Theorem 2.12, we may assume that all pebbles are distributed on the leaves of T.

Assume sω(v0) = sω(T). There exists xW\v0 satisfying dTω(v0)(x) = 1. If dT(x) ≠ 1, we can get d(x, l) > 0, and there exists a nonempty component in T\E(Tω(v0)), which is connected with x. Say T1 and b1b2≥…≥bm is the size of the maximum path partition of T1.

Case 1. D(T1) cannot move a pebble to x. |D(T1)|i=1m2bi-m, and we consider D on T\T1, |D(T\T1)| ≥ sω(T) − D(T1) ≥ sω(T\T1), and D(T\T1) is not ω-solvable, a contradiction to the minimum of |T|.

Case 2. D(T1) can move one pebble to x. It costs us at most 2b1=2dω(x,l) pebbles on T1. The left pebbles on T is not ω′-solvable (ω′ satisfies ω′(x) = ω(x) − 1, and it is unchanged for other vertices in T). From the minimum of |ω| and Corollary 3.2, we thus have |D|<sω(T)+2dω(x,l)sω(T), a contradiction to |D| = sω(T).

We may therefore assume dT(x) = 1.

We claim that D(x) = 0. Otherwise, let ω′ satisfy ω′(x) = ω(x) − 1 and ω′(v) = ω(v) for vx. Regardless of one pebble being on x, we know that |D| − 1 other pebbles cannot solve ω′. From the minimum of |ω|, we have |D|-1sω(T)-1. By Corollary 3.2, sω(T)+1sω(T), so |D| ≤ sω(T) − 1, a contradiction to |D| = sω(T), so D(x) = 0.

Assuming that x′ ~ x in T, we then delete x. Let C′(x′) = C(x′)+2C(x) and C′(v) = C(v) otherwise. Note that all pebbles are distributed on the leaves of T, so C′(x) = D(x′) − ω(x′) − 2(D(x) − ω(x)) = − ω(x′) − 2ω(x). By Lemma 2.9, D is not ω-solvable in T is equivalent to D is not ω′-solvable in T\x, where ω′(x′) = ω(x′) + 2ω(x) and ω′(v) = ω(v) for vx. By the minimum of |T|, we have |D|sω(T\x)-1, note that xv0, we have sω(T\x)=sω(T), a contradiction to |D| = sω(T). This completes the proof.

Moreover, by Theorem 1.7, we can immediately get

Corollary 3.3. Let T be a tree, and let ω be a nonnegative function on V(T), W = {vV(T):ω(v) > 0}, L = {vV(T):d(v) = 1}, then if LW,

γω(T)=maxvV(T)uV(T)ω(u)2d(u,v).

Theorem 1.4 gives a sufficient condition of a nonnegative weight function ω on V(G) for a graph G so that the ω-cover pebbling number of G is

γω(G)=maxvV(G)uV(G)ω(u)2d(u,v).

Corollary 3.3 gives a weaker sufficient condition of a nonnegative weight function ω on V(T) for a tree T so that the ω-cover pebbling number of T is

γw(T)=maxvV(T)uV(T)ω(u)2d(u,v).

Here, we explore some problems.

Problem 3.4. Give a weaker sufficient condition of a nonnegative function ω on V(G) for a graph G so that the ω-cover pebbling number of G is

γω(G)=maxvV(G)uV(G)ω(u)2d(u,v).

Problem 3.5. For a nonnegative function ω, determine the ω-cover pebbling number of more graphs, such as cycles, hypercubes, and so on.

We also give a conjecture which is similar to Graham's Conjecture.

Conjecture 3.6. Let ω1 be a nonnegative function on G and ω2 be a nonnegative function on H. The function ω on G × H is given by ω((g, h)) = ω1(g2(h), where gV(G) and hV(H), then γω(G × H) ≤ γω1(Gω2(H).

Data Availability Statement

The original contributions presented in the study are included in the article/supplementary materials, further inquiries can be directed to the corresponding authors.

Author Contributions

Z-JX provided this topic and wrote the paper. Z-JX and Z-MH solved the problem. Z-MH reviewed and edited the manuscript. All authors contributed to the article and approved the submitted version.

Funding

This research was supported by Key Projects in Natural Science Research of Anhui Provincial Department of Education (No. KJ2018A0438) to (Z-JX) and by NSFC (No. 11601002) to (Z-MH).

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Acknowledgments

This manuscript has been released as a pre-print at http://export.arxiv.org/pdf/1903.04867 (Z-JX and Z-MH). The authors are grateful for the many useful comments provided by the referees.

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Keywords: network, tree, path partition, pebbling, cover pebbling

Citation: Xia Z-J and Hong Z-M (2020) Generalization of the Cover Pebbling Number for Networks. Front. Phys. 8:197. doi: 10.3389/fphy.2020.00197

Received: 12 April 2020; Accepted: 04 May 2020;
Published: 16 June 2020.

Edited by:

Jia-Bao Liu, Anhui Jianzhu University, China

Reviewed by:

Fu-Tao Hu, Anhui University, China
Zhi Qiao, Sichuan Normal University, China
Xiang-jun Li, Yangtze University, China

Copyright © 2020 Xia and Hong. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Zheng-Jiang Xia, 120150025@aufe.edu.cn