Generalization of the cover pebbling number on trees

TL;DR

This paper extends the concept of cover pebbling numbers on trees by generalizing to nonnegative functions, unifying previous pebbling concepts and providing a comprehensive formula for trees.

Contribution

It introduces a generalized $ ext{cover pebbling number}$ for trees with nonnegative functions, broadening the scope of existing pebbling theories.

Findings

Derived a formula for the $ ext{omega}$-cover pebbling number of trees.

Unified the $t$-pebbling number and weighted cover pebbling number within a single framework.

Provided theoretical results applicable to various pebbling configurations on trees.

Abstract

A pebbling move on a graph consists of taking two pebbles off from one vertex and add 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 any vertex on $G$ regardless the original distribution of pebbles. Let $\omega$ be a positive function on $V(G)$, the $\omega$-cover pebbling number of a graph $G$ is the minimum number of pebbles so that we can reach a distribution with at least $\omega(v)$ pebbles on $v$ for all $v\in V(G)$. In this paper, we give the $\omega$-cover pebbling number of trees for nonnegative function $\omega$, which generalized the $t$-pebbling number and the traditional weighted cover pebbling number of trees.

Equations

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

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

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

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

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

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

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

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

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

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

$$s_{\omega }(v_1)-s_{\omega }(v_0)$$

$$\ge \sum _{x\in W}\omega (x)(2^{d(x,v_1)}-2^{d(x,v_0)})-2^{d(u,v_1)}$$

$$\ge \omega (u^{\prime })(2^{d(u^{\prime },v_1)}-2^{d(u^{\prime },v_0)})-2^{d(u,v_1)}$$

$$\ge 2^{d(u^{\prime },v_1)}-2^{d(u^{\prime },v_0)}-2^{d(u,v_1)}$$

$$\ge 2^{d(u^{\prime },v_1)}-\frac{2^{d(u^{\prime },v_1)}}{4}-\frac{2^{d(u^{\prime },v_1)}}{2}$$

$$=\frac{2^{d(u^{\prime },v_1)}}{4}>0.$$

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

$$s_{\omega }(T)=s_{\omega }(v_1)$$

$$=s_{{\omega }^{\prime }}(v_2)+2^{d(v,v_2)}$$

$$\ge s_{{\omega }^{\prime }}(v_2)+2^{d_{{\omega }^{\prime }}(v,l)}(\mathrm{by}\mathrm{Lemma}\text{ref}lem5)$$

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

$$s_{\omega }(T)=s_{\omega }(v_1)$$

$$=s_{{\omega }^{\prime }}(v_2)+2^{d(v,v_2)}+2^{d_{\omega }(v,l)}-2^{d_{{\omega }^{\prime }}(v,l)}$$

$$\ge s_{{\omega }^{\prime }}(v_2)+2^{d_{\omega }(v,l)}(\mathrm{by}\mathrm{Lemma}\text{ref}lem5)$$

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

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

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

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

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

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

Full text

Generalization of the cover pebbling number on trees111The research of Zheng-Jiang Xia is supported by Key Projects in Natural Science Research of Anhui Provincial Department of Education (No. KJ2018A0438). The research of Zhen-Mu Hong is supported by NSFC (No.11601002).

Zheng-Jiang Xia, Zhen-Mu Hong

School of Finance,

Anhui University of Finance & Economics,

Bengbu, Anhui, 230030, P. R. China Corresponding author, Email: xzj@mail.ustc.edu.cn (Z.-J. Xia), zmhong@mail.ustc.edu.cn (Z.-M. Hong).

Abstract: A pebbling move on a graph consists of taking two pebbles off from one vertex and add 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 any vertex on $G$ regardless the original distribution of pebbles. Let $\omega$ be a positive function on $V(G)$, the $\omega$-cover pebbling number of a graph $G$ is the minimum number of pebbles so that we can reach a distribution with at least $\omega(v)$ pebbles on $v$ for all $v\in V(G)$. In this paper, we give the $\omega$-cover pebbling number of trees for nonnegative function $\omega$, which generalized the $t$-pebbling number and the traditional weighted cover pebbling number of trees.

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

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

1 Introduction

Pebbling in graphs was first introduced by Chung[1]. For a given connected graph $G=(V,E)$, a distribution $D$ of $G$ is a projection from $V(G)$ to the nonnegative integers, $D(v)$ represents the number of pebbles on the vertex $v$, the total number of pebbles on a subset $A$ of $V$ is given by $|D(A)|=\sum_{v\in A}D(v)$, $|D|=|D(V)|$ is the size of $D$. $d(u,v)$ is the distance of $u$ and $v$, and we write $u\sim v$ if they are adjacent. $N(v)=\{u|u\sim v\}$ is the neighbor of $v$, $d(v)=|N(v)|$ is the degree of $v$, let $H$ be an induced subgraph of $G$, we use $d_{H}(v)$ to denote the degree of $v$ in $H$.

Definition 1.1

A pebbling move consists of the removal of two pebbles from a vertex and the placement of one pebble on an adjacent vertex.

Definition 1.2

The $t$-pebbling number of a vertex $v$ in $G$, denoted by $f_{t}(G,v)$, is the minimum number of pebbles that are sufficient to move $t$ pebbles to $v$ regardless of the original distribution of pebbles. *The $t$-pebbling number *of $G$, $f_{t}(G)=\max_{v\in V(G)}f_{t}(G,v)$. The pebbling number of $G$ is $f_{1}(G)$, and we denote it $f(G)$.

To determine the pebbling number of general graph is difficult. The problem of whether a distribution can reach a fixed vertex was shown to be NP-complete[4, 5]. The problem of deciding if the pebbling number of a graph $G$ is less than $k$ was shown to be $\Pi_{2}^{P}$-complete[5]. The pebbling numbers of trees[6], cycles[7], hypercubes[1], squares of cycles[10, 11] and so on have been determined. There is a conjecture given by Chung[1], which is called Graham’s Conjecture.

Conjecture 1.3

(Graham’s Conjecture)* Let $G$ and $H$ be two connected graphs, the pebbling number of the Cartesian product of $G$ and $H$ satisfies:*

[TABLE]

There are many results about Graham’s Conjecture[3], but this conjecture is still open.

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

Definition 1.4

([6]) Given a root vertex $v$ of a tree $T$, then we can view $T$ be a directed graph $\overrightarrow{T_{v}}$ with each edge directed to $v$, a path partition is a set of nonoverlapping directed paths the union of which is $\overrightarrow{T_{v}}$. A path partition is said to *majorize *another if the nonincreasing sequence of the path size majorizes that of the other (that is $(a_{1},a_{2},\ldots,a_{r})>(b_{1},b_{2},\ldots,b_{t})$ if and only if $a_{i}>b_{i}$ where $i=\min\{j:a_{j}\neq b_{j}\}$). A path partition of a tree $\overrightarrow{T_{v}}$ is said to be maximum if it majorizes all other path partitions.

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 a longest directed path $P_{1}$ in $\overrightarrow{T_{v}}$, with length $a_{1}$, then we choose a longest directed path $P_{2}$ in $\overrightarrow{T_{v}}\backslash E(P_{1})$, with length $a_{2}$, and so on.

Moews[6] found the $t$-pebbling number of trees by a path partition.

Theorem 1.5

([6])* Let $T$ be a tree, $v\in V(T)$, $(a_{1},\ldots,a_{n})$ is the size of the maximum path partition of $\overrightarrow{T_{v}}$, then*

[TABLE]

[TABLE]

Corollary 1.6

Let $T$ be a tree, $v\in V(T)$, $\alpha=(a_{1},\ldots,a_{n})$ is the size of a path partition of $\overrightarrow{T_{v}}$, $s_{\alpha}:=t2^{a_{1}}+\sum_{i=2}^{n}2^{a_{i}}-n+1$, then

[TABLE]

Proof. Let $\alpha_{0}$ be the size of the maximum path partition of $\overrightarrow{T_{v}}$, then $f_{t}(T,v)=s_{\alpha_{0}}\leq\max_{\alpha}s_{\alpha}.$

Let $P_{1},P_{2},\ldots,P_{n}$ be a path partition of $\overrightarrow{T_{v}}$, and the length of $P_{i}$ is $a_{i}$ for $1\leq i\leq n$. Note that for each $P_{i}$, assume the two endpoints $v_{i}$ and $v_{i}^{\prime}$ satisfies $d(v_{i},v)>d(v_{i}^{\prime},v)$. We put $t2^{a_{1}}-1$ pebbles on $v_{1}$, and $2^{a_{i}}-1$ pebbles on $v_{i}$ for $2\leq i\leq n$, it is clear that we cannot move $t$ pebbles on $v$ from this distribution. Thus for each $\alpha$, $s_{\alpha}-1<f_{t}(T,v)$, so $s_{\alpha}\leq f_{t}(T,v)$, so $\max_{\alpha}s_{\alpha}\leq f_{t}(T,v)$.

Definition 1.7

Let $\omega$ be a nonnegative function on $V(G)$, $D$ is a distribution on $V(G)$, we say $D$ is $\omega$-solvable (or $D$ solves $\omega$), if we can reach a distribution $D^{*}$ from $D$, by a sequence of pebbling moves, so that $D^{*}(v)\geq\omega(v)$. The* $\omega$-cover pebbling number* of $G$, denoted by $\gamma_{\omega}(G)$, is the minimum number $\gamma_{\omega}(G)$ so that every distribution $D$ with $\gamma_{\omega}(G)$ pebbles is $\omega$-solvable.

Definition 1.8

Let $\omega$ be a positive function on $V(G)$, define

[TABLE]

and

[TABLE]

The $\omega$-cover pebbling number of a graph $G$ has been determined for positive $\omega$ by [8, 9].

Theorem 1.9

([8, 9])* Let $\omega$ be a positive weight function on $V(G)$, the $\omega$-cover pebbling number of $G$ is*

[TABLE]

From Theorem 1.9, one can show

Theorem 1.10

([8, 9]) Let $\omega_{1}$ be a positive function on $G$, and $\omega_{2}$ be a positive function on $H$, the function $\omega$ on $G\times H$ is given by $\omega((g,h))=\omega_{1}(g)\omega_{2}(h)$, where $g\in V(G)$ and $h\in V(H)$, then $\gamma_{\omega}(G\times H)=\gamma_{\omega_{1}}(G)\gamma_{\omega_{2}}(H)$.

We first generalized the definition of $s_{\omega}(T)$ while $\omega$ is a nonnegative function on a tree $T$.

Definition 1.11

Given a tree $T$ and a nonnegative function $\omega$, for each vertex $v\in V(T)$, let $T_{\omega}(v)$ be the minimum subtree containing $v$ and $W:=\{u:\omega(u)>0\}$, and we give each edge in $T\backslash E(T_{\omega}(v))$ a direction towards $T_{\omega}(v)$, which is denoted by $\overrightarrow{T}\backslash E(T_{\omega}(v))$, and $(a_{1},\ldots,a_{n})$ is the size of the maximum path partition of $\overrightarrow{T}\backslash E(T_{\omega}(v))$. We define

[TABLE]

and

[TABLE]

Note that while $\omega$ is positive, then the two definitions of $s_{\omega}(T)$ are the same, so Definition 1.11 is a generalization of Definition 1.8. We generalized Theorem 1.9 while $T$ is a tree and $\omega$ is nonnegative, our main result in this paper is as follows:

Theorem 1.12

Let $T$ be a tree with a nonnegative weight function $\omega$ on $V(T)$, the $\omega$-cover pebbling number of $T$ is

[TABLE]

Theorem 1.13

Let $T$ be a tree with a nonnegative weight function $\omega$ on $V(T)$, if $|W|=1$, then $Theorem~{}\ref{thmmain}\Leftrightarrow Theorem~{}\ref{t1}$.

Proof. If $|W|=1$, assume that $\omega(v)=t$, and $\omega(u)=0$ for $u\neq v$, we will show that $f_{t}(T,v)=s_{\omega}(T)$.

Assume the size of a maximum path partition of $\vec{T}_{v}$ is $(a_{0},a_{1},\ldots,a_{n})$, and $d(v,v_{0})=a_{0}$, $P_{0}$ be the longest directed path from $v_{0}$ to $v$. Then $(a_{1},\ldots,a_{n})$ must be the size of a maximum path partition in $\vec{T}_{v}\backslash P_{0}$. So $f_{t}(T,v)=s_{\omega}(v_{0})\leq s_{\omega}(T)$.

Assume $s_{\omega}(T)=s_{\omega}(v_{1})$, and $d(v_{1},v)=a_{0}$, Let $P_{0}$ be the path connected $v_{1}$ and $v$, then $T_{\omega}(v_{1})=P_{0}$, assume $(a_{1},\ldots,a_{n})$ is the size of the maximum path partition of $T\backslash E(T_{\omega}(v))=T\backslash E(P_{0})$, so $\alpha=(a_{0},a_{1},\ldots,a_{n})$ is a path partition of $\vec{T}_{v}$, and $s_{\alpha}=s_{\omega}(v_{1})$, by Corollary 1.6, $f_{t}(T,v)\geq s_{\omega}(v_{1})=s_{\omega}(T)$, and it’s over.

Definition 1.14

([2]) 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, and each move $s\in S$ 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 next sections.

Lemma 1.15

([2], 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, reaching a distribution $D^{*}$, such that*

1. On each vertex $v$, $D^{*}(v)\geq D(v)$;

2. $T(G,S^{*})$ does not contain any directed cycles.

2 Preliminaries

Definition 2.1

Let $C$ be a generalized distribution on $G$, satisfies $C(v)$ is an integer (may be negative) for all $v\in V(G)$. A pebbling move on $G$ consists of the removal of two pebbles from a vertex $v$ (with $C(v)\geq 2$) and the placement of one pebble on an adjacent vertex.

In the following of this paper, a distribution $D$ means that $D(v)\geq 0$, and a generalized distribution $C$ means $C(v)$ is an integer for all $v\in V(G)$.

Definition 2.2

Let $\omega$ be a nonnegative function on $V(G)$, $C$ is a generalized distribution on $V(G)$, we say $C$ is $\omega$-solvable, if we can reach a distribution $C^{*}$ from $C$, by a sequence of pebbling moves, so that $C^{*}(v)\geq\omega(v)$. In particular, if $\omega(v)=0$ for all $v\in V(G)$, then we say that $C$ is [math]-solvable.

Lemma 2.3

Let $D$ be a distribution on a graph $G$ and $\omega$ be a nonnegative function on $V(G)$, $C:=D-\omega$. Then $D$ is $\omega$-solvable. $\Leftrightarrow$ $C$ is [math]-solvable.

Proof. If $C$ is [math]-solvable, then with the same sequence of pebbling moves, we can find that $D$ is $\omega$-solvable.

On the other side, if $D$ is $\omega$-solvable, by Lemma 1.15, there exist a sequence of pebbling moves $S$ reaching a distribution $D^{*}$ with $D^{*}(v)\geq\omega(v)$ and $T(G,S)$ does not contain any direct cycle. So we can give a sequence of the vertices of $G$, as $(v_{1},v_{2},\ldots,v_{n})$, so that each directed edge $v_{i}v_{j}$ in $T(G,S)$ satisfies $i<j$. Thus we can rearrangement the sequence of pebbling moves $S$ along the order $(v_{1},v_{2},\ldots,v_{n})$, that means we first choose all pebbling moves in $S$ that move pebbles off $v_{1}$, then choose all pebbling moves in $S$ that move pebbles off $v_{2}$ and so on, denote this sequence of pebbling moves $S^{\prime}$. Since no directed edges is from $v_{j}$ to $v_{i}$ for $j>i$, while we begin to move pebbles off $v_{i+1}$, then the number of pebbles left on $v_{i}$ is just $D^{*}(v_{i})(\geq\omega(v_{i})$), that means $S^{\prime}$ is a sequence of pebbling moves reaching $D^{*}-\omega\geq 0$ from $C$, thus $C$ is [math]-solvable.

Definition 2.4

Let $D$ be a distribution on a tree $T$, $\omega$ is a nonnegative function on $V(T)$, $C=D-\omega$ is called the induced generalized distribution from $D$ and $\omega$ of $T$. Let $v$ be a leaf of $T$, $u$ is its neighbor in $T$, the induced generalized distribution $C^{\prime}$ on $T\setminus v$ is given as follows: if $C(v)\geq 0$, then $C^{\prime}(u)=C(u)+\left\lfloor C(v)/2\right\rfloor$, if $C(v)<0$, then $C^{\prime}(u)=C(u)+2C(v)$, keeping $C^{\prime}(x)=C(x)$ unchanged for all $x\neq u$.

Lemma 2.5

Let $D$ be a distribution on a tree $T$, $\omega$ is a nonnegative function on $V(T)$, $C:=D-\omega$, $v$ is a leaf of $T$, $C^{\prime}$ is the induced generalized distribution from $D$ and $\omega$ of $T\backslash v$. Then $C$ is [math]-solvable in $T$. $\Leftrightarrow$ $C^{\prime}$ is [math]-solvable in $T\backslash v$.

Proof. Firstly, we assume $C$ is [math]-solvable in $T$, and there is a sequence of pebbling moves $\sigma$ reaching a distribution $C^{*}$ from $C$ with $C^{*}(x)\geq 0$ for each $x\in V(T)$.

Case $1.1$. $C(v)\geq 0$. By Lemma 1.15, we may assume that no pebble has been moved from $u$ to $v$, so at most $\left\lfloor C(v)/2\right\rfloor$ pebbles can be moved from $v$ to $u$. We may assume the first step of $\sigma$ is to move $\left\lfloor C(v)/2\right\rfloor$ pebbles from $v$ to $u$, so the left steps makes $C^{\prime}$ solves [math] on $T\backslash v$, and we are done.

Case $1.2$. $C(v)<0$. By Lemma 1.15, we may assume that no pebble has been moved from $v$ to $u$. So we may assume the last step of $\sigma$ is to move $-C(v)$ pebbles from $u$ to $v$, so the steps before it makes $C^{\prime}$ solves [math] on $T\backslash v$, and we are done.

Secondly, we assume $C^{\prime}$ is [math]-solvable in $T\backslash v$, and there is a sequence of pebbling moves $\delta$ reaching a distribution $C^{*}$ from $C^{\prime}$ with $C^{*}(x)\geq 0$ for each $x\in V(T\backslash v)$.

Case $2.1$. $C(v)\geq 0$. First, we move $\left\lfloor C(v)/2\right\rfloor$ pebbles from $v$ to $u$, and the left steps are just $\delta$, this sequence makes $C$ solves [math].

Case $2.2$. $C(v)<0$. After the sequence of pebbling moves $\delta$, we move $-C(v)$ pebbles from $u$ to $v$, this sequence makes $C$ solves [math], over.

Notations: Assume $T^{*}$ is a subtree of $T$, then $T^{*}$ can be obtained from $T$ by deleting the leaf of the subtree of $T$ (the vertex with degree one) one by one, so for each subtree $T^{*}$ of $T$, we can get an induced generalized distribution $C^{*}$. In particular, for each vertex $v\in V(T)$, 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 $\omega$ of $T_{v}$, and $\widehat{C}(v)$ to denote the induced generalized distribution of $\{v\}$.

Corollary 2.6

Let $D$ be a distribution on a tree $T$, $\omega$ is a nonnegative function on $V(T)$, and $\widehat{C}(v)$ is the induced generalized distribution from $D$ and $\omega$ of $\{v\}$. $D$ is not $\omega$-solvable.$\Leftrightarrow$ $\widehat{C}(v)<0$ for each $v\in V(T)$.

Proof. From Lemma 2.3 and Lemma 2.5, the result follows by induction.

Lemma 2.7

Let $D$ be a distribution on a tree $T$ which is not $\omega$-solvable with $|D|=\gamma_{\omega}(T)-1$, then for each vertex $x\in V(T)$ which is not a leaf of $T$, there exist a vertex $y\in N(x)$, so that $C_{x}(y)\geq 0$.

Proof. If $C_{x}(x^{\prime})<0$, for all $x^{\prime}\in N(x)$. Assume $y,z\in N(x)$ satisfies $C_{x}(z)\leq C_{x}(y)<0$, then we delete all other vertices to left $T_{1}=yxz$, and its induced generalized distribution $C_{1}$. Then $C_{1}(y)=C_{x}(y)$, $C_{1}(z)=C_{x}(z)$, and $\widehat{C}(x)=C_{1}(x)+2C_{1}(y)+2C_{1}(z)\leq-1$ by Corollary 2.6. Note that $C_{1}(x)=D(x)-w(x)+\sum_{x^{\prime}\in N(x),x^{\prime}\notin\{y,z\}}2C_{x}(x^{\prime})$. So $C_{1}(x)-D(x)\leq 0$ and $C_{1}(x)+2C_{1}(z)-D(x)\leq 0$. Now we remove $D(x)$ pebbles from $x$, and put $D(x)+1$ pebbles on $y$ to get a new distribution $D^{\prime}$ with $|D^{\prime}|=|D|+1$, the induced generalized distribution from $D^{\prime}$ and $\omega$ of $\{y\}$ is denoted by $\widehat{C^{\prime}}(y)$. Then $\widehat{C^{\prime}}(y)=(C_{1}(y)+D(x)+1)+2(C_{1}(x)+2C_{1}(z)-D(x))=(C_{1}(x)+2C_{1}(y)+2C_{1}(z))+(C_{1}(z)-C_{1}(y))+C_{1}(z)+(C_{1}(x)-D(x))+1\leq-1+0-1+0+1=-1$, so $D^{\prime}$ is not $\omega$-solvable by Corollary 2.6, a contradiction to $|D^{\prime}|=\gamma_{\omega}(T)$, and we are done.

Lemma 2.8

Let $D$ be a distribution which is not $\omega$-solvable in $T$, $x\sim y$. if $C_{x}(y)\geq 0$, then $C_{y}(x)<0$.

Proof. Let $T_{1}=xy$, and its induced generalized distribution is denoted by $C_{1}$, then $C_{1}(x)=C_{y}(x)$, $C_{1}(y)=C_{x}(y)$, if both of them are nonnegative, then $\widehat{C}(x)=C_{1}(x)+\left\lfloor C_{1}(y)/2\right\rfloor\geq 0$. By Corollary 2.6, $D$ is $\omega$-solvable, a contradiction to the condition $D$ is not $\omega$-solvable, and we are done.

Theorem 2.9

Let $\omega$ be a nonnegative function on $V(T)$, there exist a distribution $D$, which is not $\omega$-solvable with $|D|=\gamma_{\omega}(T)-1$, and all pebbles are distributed on the leaves of $T$.

Proof. We will construct such distribution as follows. Assume that $D$ is not $\omega$-solvable with $|D|=\gamma_{\omega}(T)-1$, $x$ is a vertex which is not a leaf, and $D(x)>0$, then $\widehat{C}(x)<0$ from Corollary 2.6.

From Lemma 2.7, we may assume that there exist a vertex $y\in N(x)$, with $C_{x}(y)\geq 0$, the induced generalized distribution of $T_{1}:=xy$ is denoted by $C_{1}$. Then $C_{1}(y)=C_{x}(y)\geq 0$, $\widehat{C}(x)=C_{1}(x)+\left\lfloor C_{1}(y)/2\right\rfloor<0$. Now we move all pebbles from $x$ to $y$ to get a new distribution $D^{\prime}$ with $|D^{\prime}|=|D|$, the induced generalized distribution from $D^{\prime}$ and $\omega$ of $y$ is denoted by $\widehat{C^{\prime}}(y)$. $\widehat{C^{\prime}}(y)=(C_{1}(y)+D(x))+2(C_{1}(x)-D(x))<0$, so $D^{\prime}$ is not $\omega$-solvable by Corollary 2.6.

Now we consider the new distribution $D^{\prime}$ on $T$. We use $C^{\prime}_{v}$ to denote the induced generalized distribution from $D^{\prime}$ and $\omega$ of $T_{v}$ for $v\in V(T)$. From Lemma 2.8, $C^{\prime}_{y}(x)<0$, by Lemma 2.7, there must exist $z\in N(y)$ so that $C^{\prime}_{y}(z)\geq 0$, so we can remove $D^{\prime}(y)$ pebbles from $y$ and put $D^{\prime}(y)$ pebbles on $z$ to get a new distribution $D^{\prime\prime}$ which is not $\omega$-solvable and so on, until we move the pebbles from $x$ to some leaf of $T$, and we can do the same thing to other vertex $x^{\prime}$ with $D(x^{\prime})>0$, and we are done.

3 The generalization of the cover pebbling number on trees

Assume that $s_{\omega}(v_{0})=s_{\omega}(T)$ for some $v_{0}\in V(T)$, note that $\overrightarrow{T}\backslash E(T_{\omega}(v_{0}))$ is a directed graph, we define $d_{\omega}(u,l)$ be the length of the maximal path containing $u$ in all maximum path partitions of $\overrightarrow{T}\backslash E(T_{\omega}(v_{0}))$, if $\omega$ is clear, then we use $d(u,l)$ for short (note that $d(u,l)$ maybe [math]). Let $P_{\alpha}$ be a maximal path partition of $\overrightarrow{T}\backslash E(T_{\omega}(v_{0}))$, Then $d_{\omega}(u,l)=\max_{P_{\alpha}}\{|P|:u\in P,P\in P_{\alpha}\}$.

Lemma 3.1

Assume that $s_{\omega}(v_{0})=s_{\omega}(T)$ for some $v_{0}\in V(T)$, then for each vertex $u\in V(T)$, $d(u,v_{0})\geq d(u,l)$.

Proof. If $|W|=1$, we may assume that $\omega(v)=t$, and $\omega(u)=0$ for $u\neq v$. By the proof of Theorem 1.13, we know that $f_{t}(T,v)=s_{\omega}(v_{0})$. Let $(a_{1},a_{2},\ldots,a_{n})$ be the size of the maximum path partition of $\overrightarrow{T_{v}}$. Then $d(v,v_{0})=\max_{u\in V(T)}d(v,u)=a_{1}$. Assume $P_{1}$ be the path connected $v$ and $v_{0}$. $P_{2}$ be the maximal path containing $u$ in $\overrightarrow{T_{v}}\backslash P_{1}$, and $P_{1}\cap P_{2}=v^{\prime}$, and the path connected $v$ and $v^{\prime}$ is denoted by $P_{3}$. Then the length of $P_{1}$ ($P_{2}$) is $a_{1}$ ($d(u,l)$), and $d(v^{\prime},v_{0})\leq d(u,v_{0})$. If $d(u,v_{0})<d(u,l)$, then $d(v^{\prime},v_{0})<d(u,l)$, we get a path $P_{2}\cup P_{3}$ with length $a_{1}-d(v^{\prime},v_{0})+d(u,l)>a_{1}$, a contradiction to the maximum of $a_{1}$, thus $d(u,v_{0})\geq d(u,l)$.

If $|W|\geq 2$. We only need to show it while $u\in V(T_{\omega}(v_{0}))$.

If $d(u,v_{0})<d(u,l)$ for some $u\in V(T_{\omega}(v_{0}))$, let $u^{\prime}$ be some vertex in $W\backslash u$. Then there exist some leaf $v_{1}$ in $\overrightarrow{T}\backslash E(T_{\omega}(v_{0}))$ so that $d(u,v_{0})<d(u,v_{1})$, we will show that $s_{\omega}(v_{1})>s_{\omega}(v_{0})$.

Let $P$ be the path connected $v_{1}$ and $u$, then we know the inner vertices of $P$ do not belong to $W$, so for each $x\in W\backslash u$, $d(x,v_{1})>d(u,v_{1})$ and $d(x,v_{1})-d(x,v_{0})\geq d(u,v_{1})-d(u,v_{0})+1\geq 2$.

Note that $\overrightarrow{T}\backslash E(T_{\omega}(v_{0}))\backslash P\subseteq\overrightarrow{T}\backslash E(T_{\omega}(v_{1}))$. So

[TABLE]

Which is a contradiction to $s_{\omega}(v_{0})=s_{\omega}(T)$, and we are done.

Corollary 3.2

Let $\omega$ be a nonnegative function in $V(T)$, for some $v\in W$, $\omega^{\prime}$ be a nonnegative function satisfies $\omega^{\prime}(v)=\omega(v)-1$, $\omega^{\prime}(u)=\omega(u)$ for other vertices in $T$, then

[TABLE]

Proof. Assume that there exist $v_{1}$ and $v_{2}$, so that $s_{\omega}(v_{1})=s_{\omega}(T)$ and $s_{\omega^{\prime}}(v_{2})=s_{\omega^{\prime}}(T)$.

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

[TABLE]

If $\omega(v)=1$, the difference between $\overrightarrow{T}\backslash T_{\omega}(v_{1})$ and $\overrightarrow{T}\backslash T_{\omega^{\prime}}(v_{2})$ is just the length of the maximal path containing $v$, so we have

[TABLE]

Theorem 3.3

Let $\omega$ be a nonnegative function on $V(T)$, the $\omega$-cover pebbling number of $T$ is

[TABLE]

Proof. The lower bound holds clearly, for we put $2^{a_{i}}-1$ pebbles on the leaf of each path for $1\leq i\leq n$ (then no pebble can be moved to $T_{\omega}(v)$), and $\sum_{u\in S}w(u)2^{d(u,v)}-1$ pebbles on $v$, obviously it is not $\omega$-solvable, and we are done.

For the upper bound, it holds if $|\omega|=1$ or $|W|=1$ by the proof of Theorem 1.13, also it holds for $|T|\leq 2$ by Theorem 1.5 and Theorem 1.9. So we may assume that $|\omega|\geq 2$, $|W|\geq 2$ and $|T|\geq 3$.

If the result is false for some $T$ and $\omega$, then we choose one counterexample $T$ and its weight $\omega$, so that $|T|$ and $|\omega|$ are both minimal, that means the upper bound holds for $T^{\prime}$ and its weight $\omega^{\prime}$ if $|T^{\prime}|<|T|$ or $|\omega^{\prime}|<|\omega|$.

Let $D$ be a distribution on $T$ which is not $\omega$-solvable with $s_{\omega}(T)$ pebbles, by Theorem 2.9, we may assume that all pebbles are distributed on the leaves of $T$.

Let $s_{\omega}(v_{0})=s_{\omega}(T)$ , there exist $x\in W\backslash v_{0}$ satisfies $d_{T_{\omega}(v_{0})}(x)=1$, then if $d_{T}(x)\neq 1$, we can get $d(x,l)>0$, and there exist a nonempty connected component in $T\backslash E(T_{\omega}(v_{0}))$ which is connected with $x$, say $T_{1}$, and $b_{1}\geq b_{2}\geq\ldots\geq b_{m}$ is the size of the maximum path partition of $T_{1}$.

Case $1$. $D(T_{1})$ cannot move a pebble to $x$, then $|D(T_{1})|\leq\sum_{i=1}^{m}2^{b_{i}}-m$, then we consider $D$ on $T\backslash T_{1}$, $|D(T\backslash T_{1})|\geq s_{\omega}(T)-D(T_{1})\geq s_{\omega}(T\backslash T_{1})$, and also $D(T\backslash T_{1})$ is not $\omega$-solvable, a contradiction to the minimum of $|T|$.

Case $2$. $D(T_{1})$ can move one pebble to $x$, then it cost us at most $2^{b_{1}}=2^{d_{\omega}(x,l)}$ pebbles on $T_{1}$, the left pebbles on $T$ is not $\omega^{\prime}$-solvable ($\omega^{\prime}$ satisfies $\omega^{\prime}(x)=\omega(x)-1$, and unchanged for other vertices in $T$), so from the minimum of $|\omega|$ and Corollary 3.2, we have $|D|<s_{\omega^{\prime}}(T)+2^{d_{\omega}(x,l)}\leq s_{\omega}(T)$, a contradiction to $|D|=s_{\omega}(T)$.

So we may assume $d_{T}(x)=1$.

We claim that $D(x)=0$. Otherwise, let $\omega^{\prime}$ satisfies $\omega^{\prime}(x)=\omega(x)-1$ and $\omega^{\prime}(v)=\omega(v)$ for $v\neq x$. Ignore one pebble already on $x$, we know that $|D|-1$ other pebbles cannot solve $\omega^{\prime}$, from the minimum of $|\omega|$, we have $|D|-1\leq s_{\omega^{\prime}}(T)-1$. By Corollary 3.2, $s_{\omega^{\prime}}(T)+1\leq s_{\omega}(T)$, so $|D|\leq s_{\omega}(T)-1$, a contradiction to $|D|=s_{\omega}(T)$, so $D(x)=0$.

Assume that $x^{\prime}\sim x$ in $T$, then we delete $x$, let $C^{\prime}(x^{\prime})=C(x^{\prime})+2C(x)$, and $C^{\prime}(v)=C(v)$ otherwise. Note that all pebbles are distributed on the leaves of $T$, so $C^{\prime}(x)=D(x^{\prime})-\omega(x^{\prime})-2(D(x)-\omega(x))=-\omega(x^{\prime})-2\omega(x)$. By Lemma 2.5, $D$ is not $\omega$-solvable in $T$ is equivalent to $D$ is not $\omega^{\prime}$-solvable in $T\backslash x$, where $\omega^{\prime}(x^{\prime})=\omega(x^{\prime})+2\omega(x)$ and $\omega^{\prime}(v)=\omega(v)$ for $v\neq x$. By the minimum of $|T|$, we have $|D|\leq s_{\omega^{\prime}}(T\backslash x)-1$, note that $x\neq v_{0}$, we have $s_{\omega^{\prime}}(T\backslash x)=s_{\omega}(T)$, a contradiction to $|D|=s_{\omega}(T)$, and we are done.

Moreover, from Theorem 3.3, we can immediately get

Corollary 3.4

Let $T$ be a tree, $\omega$ be a nonnegative function on $V(T)$, $W=\{v\in V(T):\omega(v)>0\}$, $L=\{v\in V(T):d(v)=1\}$, then if $L\subseteq W$,

[TABLE]

Theorem 1.9 gives a sufficient condition of a nonnegative weight function $\omega$ on $V(G)$ for a graph $G$ so that the $\omega$-cover pebbling number of $G$ is

[TABLE]

Corollary 3.4 gives a weaker sufficient condition of a nonnegative weight function $\omega$ on $V(T)$ for a tree $T$ so that the $\omega$-cover pebbling number of $T$ is

[TABLE]

Here we give some problems.

Problem 3.5

Give a weaker sufficient condition of a nonnegative function $\omega$ on $V(G)$ for a graph $G$ so that the $\omega$-cover pebbling number of $G$ is

[TABLE]

Problem 3.6

For a nonnegative function $\omega$, determine the $\omega$-cover pebbling number of more graphs, such as cycles, hypercubes and so on.

Also we give a conjecture which is similar to Graham’s Conjecture.

Conjecture 3.7

Let $\omega_{1}$ be a nonnegative function on $G$, and $\omega_{2}$ be a nonnegative function on $H$, the function $\omega$ on $G\times H$ is given by $\omega((g,h))=\omega_{1}(g)\omega_{2}(h)$, where $g\in V(G)$ and $h\in V(H)$, then $\gamma_{\omega}(G\times H)\leq\gamma_{\omega_{1}}(G)\gamma_{\omega_{2}}(H)$.

Acknowledgments. The authors are grateful for the many useful comments provided referees.