Generalization of the Cover Pebbling Number for Networks

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 v ∈ V(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 d_H(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, $\left|D\right|=\sum _{v\in V(D)}D(v)$ 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 v ∈ V(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 f_t(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), v ∈ E(H)} and {((u, x), (u, y))|u ∈ V(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 v ∈ V(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 ${\Pi }_2^P$-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:

$f(G\times H)\le f(G)f(H).$

There are many results about Graham's Conjecture [6–10], 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 v ∈ V(G). The ω-cover pebbling number of G, denoted by γ_ω(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

$s_{\omega }(v)={\displaystyle \sum _{u\in V(G)}}\omega (u)2^{d(u,v)},$

and

$s_{\omega }(G)={\displaystyle \underset{v\in V(G)}{max}}{s}_{\omega }(v).$

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

${\gamma }_{\omega }(G)=s_{\omega }(G).$

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(g)ω₂(h), where g ∈ V(G) and h ∈ V(H), then γ_ω(G × H) = γ_ω1(G)γ_ω2(H).

We first generalize the definition of s_ω(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 v ∈ V(T), and let T_ω(v) be the minimum subtree of T containing v and W: = {u:ω(u) > 0}. We give each edge in T\E(T_ω(v)) a direction toward T_ω(v) to get a directed graph, which is denoted by $\vec{T}\backslash E(T_{\omega }(v))$, and (a₁, …, a_n) is the size of the maximum path partition of $\vec{T}\backslash E(T_{\omega }(v))$. We define

$s_{\omega }(v)={\displaystyle \sum _{u\in W}}\omega (u)2^{d(u,v)}+{\displaystyle \sum _{i=1}^n}{2}^{a_i}-n.$

and

$s_{\omega }(T)={\displaystyle \underset{v\in V(T)}{max}}{s}_{\omega }(v).$

Note that while ω is positive, then the two definitions of s_ω(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

${\gamma }_{\omega }(T)=s_{\omega }(T).$

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 u ≠ v. We will show that f_t(T, v) = s_ω(T).

Assume the size of a maximum path partition of ${\overrightarrow{T}}_v$ is (a₀, a₁, …, a_n), and d(v, v₀) = a₀, P₀ is the longest directed path from v₀ to v. Then (a₁, …, a_n) must be the size of a maximum path partition in ${\overrightarrow{T}}_v\backslash P_0$. So f_t(T, v) = s_ω(v₀) ≤ s_ω(T).

Assume s_ω(T) = s_ω(v₁), and d(v₁, v) = a₀. Let P₀ be the path connected v₁ and v, then T_ω(v₁) = P₀; assume (a₁, …, a_n) is the size of the maximum path partition of T\E(T_ω(v)) = T\E(P₀), so α = (a₀, a₁, …, a_n) is a path partition of ${\overrightarrow{T}}_v$, and s_α = s_ω(v₁) by Corollary 2.3 and f_t(T, v) ≥ s_ω(v₁) = s_ω(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 s ∈ S 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 $\overrightarrow{T_v}$ with edges directed toward v. A path partition is a set of nonoverlapping directed paths in which the union is $\overrightarrow{T_v}$. 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, …, a_r) > (b₁, b₂, …, b_t) if and only if a_i > b_i, where i = min{j:a_j ≠ b_j}). A path partition of a tree $\overrightarrow{T_v}$ 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 $\overrightarrow{T_v}$, with length a₁. Then, we choose the longest directed path P₂ in $\overrightarrow{T_v}\backslash E(P_1)$, 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, v ∈ V(T), and (a₁, …, a_n) be the size of the maximum path partition of $\overrightarrow{T_v}$. Then,

$f_t(T,v)=t{2}^{a_1}+{\displaystyle \sum _{i=2}^n}{2}^{a_i}-n+1,$

$f_t(T)={\displaystyle \underset{v\in V(T)}{max}}{f}_t(T,v).$

Corollary 2.3. Let T be a tree, v ∈ V(T), and α = (a₁, …, a_n) be the size of a path partition of $\overrightarrow{T_v}$, $s_{\alpha }:=t{2}^{a_1}+\sum _{i=2}^n{2}^{a_i}-n+1$. Then,

$f_t(T,v)={\displaystyle \underset{\alpha }{max}}{s}_{\alpha }.$

Proof. Let α₀ be the size of the maximum path partition of $\overrightarrow{T_v}$. Then, $f_t(T,v)=s_{{\alpha }_0}\le \underset{\alpha }{max}{s}_{\alpha }.$

Assume P₁, P₂, …, P_n is a path partition of $\overrightarrow{T_v}$, and the length of P_i is a_i for 1 ≤ i ≤ n. Note that for each P_i we should assume the two endpoints v_i and $v_i^{\prime }$ satisfy $d(v_i,v)>d(v_i^{\prime },v)$. We put $t{2}^{a_1}-1$ pebbles on v₁ and $2^{a_i}-1$ pebbles on v_i for 2 ≤ i ≤ n; it is clear that t pebbles cannot be moved to v from this distribution. Thus, for each α, s_α − 1 < f_t(T, v), so s_α ≤ f_t(T, v) so $\underset{\alpha }{max}{s}_{\alpha }\le f_t(T,v)$. ❚

Definition 2.4. Let C be a generalized distribution on G, where C(v) is an integer (may be negative) for all v ∈ V(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 v ∈ V(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, …, S_k), where S_i is a pebbling move for 1 ≤ i ≤ k, and the pebbling move S_i is in front of S_j if 1 ≤ i < j ≤ k. We say the pebbling sequence S executable, if S_i is executable for 1 ≤ i ≤ k.

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 v ∈ V(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, …, v_n), so that each directed edge v_iv_j in T(G, S) satisfies i < j. We can thus rearrange the sequence of pebbling moves S along the order (v₁, v₂, …, v_n). 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 v ∈ V(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 v_i, we first move α_i pebbles from other vertices to v_i and then remove β_i pebbles from v_i.

We only need to show that, for each v_i ∈ V(G), the sequence of pebbling moves that removes β_i pebbles from v_i in S′, denoted by σ_i, is executable. We use induction on i. If i = 1, and we can then get $D(v_1)-{\beta }_1=D^{*}(v_1)\ge \omega (v_1)\Rightarrow D(v_1)-\omega (v_1)\ge {\beta }_1\Rightarrow C(v_1)\ge {\beta }_1$, and so σ₁ is executable.

Assume σ_h is executable for h < i. By induction, the pebbling sequence that moves α_i pebbles to v_i is executable. Moreover, we can get $D(v_i)+{\alpha }_i-{\beta }_i=D^{*}(v_i)\Rightarrow$ $D(v_i)+{\alpha }_i-\omega (v_i)-{\beta }_i=$ $D^{*}(v_i)-\omega (v_i)\ge 0\Rightarrow D(v_i)-\omega (v_i)+$ ${\alpha }_i\ge {\beta }_i\Rightarrow C(v_i)+{\alpha }_i\ge {\beta }_i$. Thus σ_i 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 x ≠ u.

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 x ∈ V(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 x ∈ V(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 v ∈ V(T), let T_v be a subtree containing v and all of its neighbors. We use C_v to denote the induced generalized distribution from D and ω of T_v and $\hat{C}(v)$ 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 $\hat{C}(v)$ be the induced generalized distribution from D and ω of {v}. D is not ω-solvable is equivalent to $\hat{C}(v)<0$ for each v ∈ V(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| = γ_ω(T) − 1. For each vertex x ∈ V(T), which is not a leaf of T, there exists a vertex y ∈ N(x), so that C_x(y) ≥ 0.

Proof. If $C_x(x^{\prime })<0$, for all x′ ∈ N(x), assume y, z ∈ N(x) with C_x(z) ≤ C_x(y) < 0. We delete all other vertices to the left T₁ = yxz and its induced generalized distribution C₁. Then, C₁(y) = C_x(y), C₁(z) = C_x(z) and $\hat{C}(x)=C_1(x)+2C_1(y)+2C_1(z)\le -1$ by Corollary 2.10. Note that $C_1(x)=D(x)-w(x)+\sum _{x^{\prime }\in N(x),x^{\prime }\notin \left\{y,z\right\}}2C_x(x^{\prime })$. 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 $\widehat{C^{\prime }}(y)$. Then, $\widehat{C^{\prime }}(y)=(C_1(y)+D(x)+1)+$ $2(C_1(x)+2C_1(z)-D(x))=$ $(C_1(x)+2C_1(y)+2C_1(z))+(C_1(z)-C_1(y))$ $+C_1(z)+$ $(C_1(x)-D(x))+1\le -1+0-1+0+1=-1$, and so D′ is not ω-solvable by Corollary 2.10, a contradiction to $\left|D^{\prime }\right|={\gamma }_{\omega }(T)$, 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| = γ_ω(T) − 1. All pebbles are then distributed on the leaves of T.

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

If not, that means for all v ∈ N(x), C_x(v) ≥ 0. Note that D(x) > 0, and thus $\hat{C}(x)=D(x)+\sum _{v\in N(x)}\lfloor C_x(v)/2\rfloor >0$. By Corollary 2.10, D is ω-solvable, a contradiction. Thus, there exists a vertex z ∈ N(x) with C_x(z) < 0.

Let T₁ = yxz be the subtree of T, with induced generalized distribution C₁. Then, C₁(z) = C_x(z), C₁(y) = C_x(y), and $\hat{C}(x)=C_1(x)+\lfloor C_1(y)/2\rfloor +2C_1(z)<0$.

Now, consider the new distribution D^*, with D^*(y) = D(y) + D(x)+1, D^*(x) = 0, and D^*(v) = D(v); $\left|D^{*}\right|={\gamma }_{\omega }(T)$. The induced generalized distribution from D^* and ω of {x} is given by $\widehat{C^{*}}(x)=(C_1(x)-D(x))+\lfloor (C_1(y)+D(x)+1)/2\rfloor +2C_1(z)$.

If D(x) = 1, then $\widehat{C^{*}}(x)=C_1(x)+\lfloor C_1(y)/2\rfloor +2C_1(z)=\hat{C}(x)<0$;

If D(x) ≥ 2, then $\widehat{C^{*}}(x)\le C_1(x)-D(x)+\lfloor C_1(y)/2\rfloor +D(x)/2+1$ $+2C_1(z)=\hat{C}(x)+1-D(x)/2\le \hat{C}(x)<0$.

By Corollary 2.10, D^* is not ω-solvable, a contradiction to $\left|D^{*}\right|={\gamma }_{\omega }(T)$. This completes the proof. ❚

From Theorem 2.12, for a given integer p with p < γ_ω(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 s_ω(v₀) = s_ω(T) for some v₀ ∈ V(T); it should be noted that $\vec{T}\backslash E(T_{\omega }(v_0))$ is a directed graph. We define d_ω(u, l) to be the length of the maximal path containing u in all maximum path partitions of $\vec{T}\backslash E(T_{\omega }(v_0))$. If ω is clear, then we use d(u, l) for short (note that d(u, l) maybe 0). Let P_α be a maximal path partition of $\vec{T}\backslash E(T_{\omega }(v_0))$; then, $d_{\omega }(u,l)=\underset{P_{\alpha }}{max}\left\{\left|P\right|:u\in P,P\in P_{\alpha }\right\}$.

Lemma 3.1. Assume that s_ω(v₀) = s_ω(T) for some v₀ ∈ V(T); then for each vertex u ∈ V(T) and d(u, v₀) ≥ d(u, l).

Proof. Assume u, v ∈ V(T). There is exactly one subpath of T with endpoints u and v, and we denote this path by P_uv. We thus have P_uv = P_vu.

If |W| = 1, we may assume that ω(v) = t, and ω(u) = 0 for u ≠ v. By the proof of Theorem 1.8, we know that f_t(T, v) = s_ω(v₀). Let (a₁, a₂, …, a_n) be the size of the maximum path partition of $\overrightarrow{T_v}$. Then $d(v,v_0)=\underset{u\in V(T)}{max}d(v,u)=a_1$. Assume P₁ is the maximal path containing u in $\overrightarrow{T_v}\backslash P_{v_0,v}$, and P₁∩P_v0v = v₁. The length of P_v0v (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 P₁∪P_v1v with length a₁ − d(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 u ∈ V(T_ω(v₀)).

If d(u, v₀) < d(u, l) for some u ∈ V(T_ω(v₀)), there exists a leaf v₁ in $\vec{T}\backslash E(T_{\omega }(v_0))$ so that d(u, l) = d(u, v₁), and we will show that s_ω(v₁) > s_ω(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 w₁∈TC(v₀) ∩ W, then d(w₁, v₁) ≥ d(u, v₁) + 1 and

$d(w_1,v_1)-d(w_1,v_0)\ge d(u,v_1)-d(u,v_0)+2\ge 3.$

Note that $\vec{T}\backslash E(T_{\omega }(v_0)\cup P_{v_{1}u})\subseteq \vec{T}\backslash E(T_{\omega }(v_1))$. So

$\begin{array}{l}{s}_{\omega }(v_1)-s_{\omega }(v_0)\\ \ge {\displaystyle \sum _{x\in W}}\omega (x)(2^{d(x,v_1)}-2^{d(x,v_0)})-2^{d(u,v_1)}\\ \ge \omega (w_1)(2^{d(w_1,v_1)}-2^{d(w_1,v_0)})-2^{d(u,v_1)}\\ \ge 2^{d(w_1,v_1)}-2^{d(w_1,v_0)}-2^{d(u,v_1)}\\ \ge 2^{d(w_1,v_1)}-\frac{2^{d(w_1,v_1)}}{8}-\frac{2^{d(w_1,v_1)}}{2}\\ =\frac{3·2^{d(w_1,v_1)}}{8}>0.\end{array}$

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

Case 2. TC(v₀) ∩ W = ∅.

Let ${\tau }_{\omega }(v)=\sum _{x\in W}\omega (x)2^{d(x,v)}$. If so, then ${\tau }_{\omega }(v_0)=2^{d(u,v_0)}{\tau }_{\omega }(u)$, and ${\tau }_{\omega }(v_1)=2^{d(u,v_1)}{\tau }_{\omega }(u)$. For |W| ≥ 2, ${\tau }_{\omega }(u)\ge 2^0+2^1=3$.

Note that $\vec{T}\backslash E(T_{\omega }(v_0)\cup P_{v_{1}u})\subseteq \vec{T}\backslash E(T_{\omega }(v_1))$. So

$\begin{array}{l}{s}_{\omega }(v_1)-s_{\omega }(v_0)\\ \ge 2^{d(u,v_1)}{\tau }_{\omega }(u)-2^{d(u,v_0)}{\tau }_{\omega }(u)-2^{d(u,v_1)}\\ ={\tau }_{\omega }(u)(2^{d(u,v_1)}-2^{d(u,v_0)})-2^{d(u,v_1)}\\ \ge 3(2^{d(u,v_1)}-2^{d(u,v_0)})-2^{d(u,v_1)}\\ \ge 3(2^{d(u,v_1)}-\frac{2^{d(u,v_1)}}{2})-2^{d(u,v_1)}\\ =\frac{2^{d(u,v_1)}}{2}>0.\end{array}$

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

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

$s_{\omega }(T)\ge s_{{\omega }^{\prime }}(T)+2^{d_{\omega }(v,l)}.$

Proof. Assume that there exist v₁ and v₂, so that s_ω(v₁) = s_ω(T) and $s_{{\omega }^{\prime }}(v_2)=s_{{\omega }^{\prime }}(T)$.

By the definition of s_ω(v), if ω(v) ≥ 2, then $d_{\omega }(v,l)=d_{{\omega }^{\prime }}(v,l)$, we have

$\begin{array}{l}{s}_{\omega }(T)=s_{\omega }(v_1)\ge s_{\omega }(v_2)\\ =s_{{\omega }^{\prime }}(v_2)+2^{d(v,v_2)}\\ \ge s_{{\omega }^{\prime }}(v_2)+2^{d_{{\omega }^{\prime }}(v,l)}(\text{by Lemma }3.1)\\ =s_{{\omega }^{\prime }}(T)+2^{d_{\omega }(v,l)}.\end{array}$

If ω(v) = 1, the difference between $\vec{T}\backslash T_{\omega }(v_1)$ and $\vec{T}\backslash T_{{\omega }^{\prime }}(v_2)$ is just the length of the maximal path containing v, we have

$\begin{array}{l}{s}_{\omega }(T)=s_{\omega }(v_1)\ge s_{\omega }(v_2)\\ =s_{{\omega }^{\prime }}(v_2)+2^{d(v,v_2)}+2^{d_{\omega }(v,l)}-2^{d_{{\omega }^{\prime }}(v,l)}\\ \ge s_{{\omega }^{\prime }}(v_2)+2^{d_{\omega }(v,l)}(\text{by Lemma }3.1)\\ =s_{{\omega }^{\prime }}(T)+2^{d_{\omega }(v,l)}.\end{array}$

The proof of Theorem 1.7:

The lower bound holds clearly, as we put $2^{a_i}-1$ pebbles on the leaf of each path for 1 ≤ i ≤ n (no pebble can then be moved to T_ω(v)), and $\sum _{u\in S}w(u)2^{d(u,v)}-1$ 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 s_ω(T). By Theorem 2.12, we may assume that all pebbles are distributed on the leaves of T.

Assume s_ω(v₀) = s_ω(T). There exists x ∈ W\v₀ satisfying d_Tω(v0)(x) = 1. If d_T(x) ≠ 1, we can get d(x, l) > 0, and there exists a nonempty component in T\E(T_ω(v₀)), which is connected with x. Say T₁ and b₁ ≥ b₂≥…≥b_m is the size of the maximum path partition of T₁.

Case 1. D(T₁) cannot move a pebble to x. $\left|D(T_1)\right|\le \sum _{i=1}^m{2}^{b_i}-m$, and we consider D on T\T₁, |D(T\T₁)| ≥ s_ω(T) − D(T₁) ≥ s_ω(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 $2^{b_1}=2^{d_{\omega }(x,l)}$ 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 $\left|D\right|<s_{{\omega }^{\prime }}(T)+2^{d_{\omega }(x,l)}\le s_{\omega }(T)$, a contradiction to |D| = s_ω(T).

We may therefore assume d_T(x) = 1.

We claim that D(x) = 0. Otherwise, let ω′ satisfy ω′(x) = ω(x) − 1 and ω′(v) = ω(v) for v ≠ x. Regardless of one pebble being on x, we know that |D| − 1 other pebbles cannot solve ω′. From the minimum of |ω|, we have $\left|D\right|-1\le s_{{\omega }^{\prime }}(T)-1$. By Corollary 3.2, $s_{{\omega }^{\prime }}(T)+1\le s_{\omega }(T)$, so |D| ≤ s_ω(T) − 1, a contradiction to |D| = s_ω(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 v ≠ x. By the minimum of |T|, we have $\left|D\right|\le s_{{\omega }^{\prime }}(T\backslash x)-1$, note that x≠v₀, we have $s_{{\omega }^{\prime }}(T\backslash x)=s_{\omega }(T)$, a contradiction to |D| = s_ω(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 = {v ∈ V(T):ω(v) > 0}, L = {v ∈ V(T):d(v) = 1}, then if L ⊆ W,

${\gamma }_{\omega }(T)={\displaystyle \underset{v\in V(T)}{max}}{\displaystyle \sum _{u\in V(T)}}\omega (u)2^{d(u,v)}.$

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

${\gamma }_{\omega }(G)={\displaystyle \underset{v\in V(G)}{max}}{\displaystyle \sum _{u\in V(G)}}\omega (u)2^{d(u,v)}.$

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

${\gamma }_w(T)={\displaystyle \underset{v\in V(T)}{max}}{\displaystyle \sum _{u\in V(T)}}\omega (u)2^{d(u,v)}.$

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

${\gamma }_{\omega }(G)={\displaystyle \underset{v\in V(G)}{max}}{\displaystyle \sum _{u\in V(G)}}\omega (u)2^{d(u,v)}.$

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(g)ω₂(h), where g ∈ V(G) and h ∈ V(H), then γ_ω(G × H) ≤ γ_ω1(G)γ_ω2(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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