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Graph SequencesFor this section we will fix notation as follows. The vertex set for any graph on $N$ vertices will be taken to be $\{v_i | i \in [N]\}$, where $[N]=\{0,\ldots,N-1\}$. That way, any configuration $C : V(G_n) \rightarrow N = \{0,1,2,\ldots \}$ is independent of $G_n$. (Here we make the distinction that $n$ is the index of the graph $G_n$ in a sequence $\mathbf{G}=(G_1,\ldots,G_n,\ldots)$, whereas $N=N(G_n)$ denotes its number of vertices.) Let ${\cal K}=(K_1,\ldots,K_n,\ldots)$, denote the sequence of complete graphs, ${\cal P}=(P_1,\ldots,P_n,\ldots)$, the sequence of paths, and ${\cal Q}=(Q^1,\ldots,Q^n,\ldots)$, the sequence of $n$-dimensional cubes. Let $C_n : [N] \rightarrow \mathbb{N}$ denote a configuration on $N=N(G_n)$ vertices.

ThresholdsLet $h : \mathbb{N} \rightarrow \mathbb{N}$ and for fixed $N$ consider the probability space $X_N$ of all configurations $C_n$ of size $h=h(N)$. We denote by $P_N^+$ the probability that $C_n$ is $G_n$-solvable and let $t : \mathbb{N} \rightarrow \mathbb{N}$. We say that $t$ is a pebbling threshold for ${\cal G}$ and write $\tau({\cal G}) = \Theta(t)$ if $P_N^+ \rightarrow 0$. whenever $h(N) \ll t(N)$ and $P_N^+ \rightarrow 1$ whenever $h(N) \gg t(N)$. The existence of such thresholds was established in [BBCH].Pebbling Threshold Existence Theorem. Every graph sequence ${\cal G}$ has a pebbling threshold $\tau({\cal G})$.

Cliques and PathsThe first threshold result is found in [Cla]. The result is merely an unlabeled version of the so-called "Birthday problem", in which one finds the probability that $2$ of $t$ people share the same birthday, assuming $N$ days in a year.Clique Threshold. $\tau({\cal K})=\Theta(\sqrt{N})$.The same threshold applies to the sequence of stars ${\cal S}=(K_{1,1},\ldots,K_{1,n},\ldots)$.It was discovered in [CEHK] that every graph sequence ${\cal G}$ satisfies $\tau({\cal K}) < \sim \tau({\cal G}) < \sim \tau({\cal P})$, where $A < \sim B$ is meant to signify that $a \in O(b)$ for every $a \in A, b \in B$. The authors also discovered that $\tau({\cal G}) \subset O(N)$. when ${\cal G}$ is a sequence of graphs of bounded diameter, that $\tau({\cal Q}) \subset O(N)$ and that $\tau({\cal P}) \subset \Omega(N)$. Surprisingly, the threshold for the sequence of paths has not been determined. The lower bound of $\Omega(n)$ found in [CEHK] was improved in [BBCH] to $\tau({\cal P}) \subset \Omega(N2^{c \sqrt{\log N}})$ for every $c < 1/ \sqrt{2}$, while the upper bound of $\tau({\cal P}) \subset \Omega(N2^{2 \sqrt{\log N}})$ found in [BBCH] was improved in [GJSW] to $\tau({\cal P}) \subset \Omega(N2^{c \sqrt{\log N}})$ for every $c > 1$. Finally the lower bound was tightened in [CzHu2] to nearly match the upper bound.

Path Threshold Bounds. For any constant $c > 1$, we have $\tau({\cal P}) \subset \Omega(N2^{M/c}) \cap O(N2^{cM})$, where $M=\sqrt{\log N}$.

DensityIt is natural to think that any given function could be the threshold for some graph sequence.Threshold Density Conjecture. Let $t_1,t_2$ be any functions satisfying $\tau({\cal K}) < \sim t_1 \ll \tau({\cal P})$. Then there is some graph sequence ${\cal G}$ such that $t_1 < \sim \tau({\cal G}) < \sim t_2$.This conjecture was proven in [CzHu2] in the case that $\tau({\cal P})$ is replaced by $\Theta(N)$. In fact, the family of fuses covers this whole range. What behavior lives above $\Theta(N)$ remains unknown.

ProductsIt is interesting to consider a pebbling threshold version of Graham's Conjecture. Given graph sequences ${\cal F}$ and ${\cal G}$, define the sequence ${\cal H}={\cal F} \times {\cal G} = (F_1 \times G_1, \ldots, F_n \times G_n, \ldots)$. Suppose that $f(R) \in \tau({\cal F})$. $g(S) \in \tau({\cal G})$, and $h(T) \in \tau({\cal H})$, where $R=N(F_n)$, $S=N(G_n)$, and $T=N(H_n)=RS$.Threshold Product Question. Is it true that, for ${\cal F}$, ${\cal G}$, and ${\cal H}$ as defined above, we have $h(T) \in O(f(R)g(S))$?In particular, one can define the sequence of graphs ${\cal G}^k$ in the obvious way. In [CzHu3] one finds tight enough bounds on $\tau({\cal P}^k)$ to show that the answer to this question is yes for ${\cal F}={\cal P}^i$ and ${\cal G}={\cal P}^j$.Grid Threshold Bounds. Let $N=n^d$ be the number of vertices of ${\cal P}_n^d$ and let $M=(\log N)^{1/(d+1)}$. Then $\tau({\cal P}^d) \subset \Omega(N2^{cM}) \cap O(N2^{c'M})$ for all $c < 2^{-d/(d+1)}$ and $c' > d+1$.Another important instance is ${\cal H} = {\cal K}^2$. Boyle [Boyl] proved that $\tau({\cal K}^2) \in O(N^{3/4})$. This was improved in [BeHu], answering the Threshold Product Question affirmatively for squares of complete graphs.

Square Clique Threshold. For ${\cal K}^2=(K_1^2, \ldots, K_n^2, \ldots)$ we have $\tau({\cal K}^2)=\Theta(N^{1/2})$.This result is interesting because, by squaring, the graphs become fairly sparse, and yet their structure maintains the low pebbling threshold. The proof of the result tied the behavior of pebbling in ${\cal K}_n^2$ to the existence of large components in various models of random complete bipartite graphs.Another interesting related sequence to consider is ${\cal P}_l = (P_l^1, \ldots, P_l^n, \ldots).$ When $l=2$ we have ${\cal P}_2={\cal Q}$, and the best result to date is the following theorem of [CzWa], (obtained independently by Alon).

Cube Threshold Bounds. For the sequence of cubes we have $\tau({\cal Q}) \in \Omega(N^{1 - \varepsilon}) \cap O(N/(\log \log N)^{1 - \varepsilon})$ for all $\varepsilon > 0$.Let $l=l(n)$ and $d=d(n)$ and denote by ${\cal P}_l^d$ the graph sequence $(P_l^d)_{n > 0}$ where $P_l^d=(P_l)^d$. Most likely, fixed $l$ yields similar behavior to the Cube Threshold Bounds.Grid Threshold Conjecture. For fixed $l$ we have $\tau({\cal P}_l^n) \in o(N)$.In contrast, the Grid Threshold Bounds show that $\tau({\cal P}^d) \in \Omega(N)$ for fixed $d$. Thus it is reasonable to believe that there should be some relationship between the two functions $l=l(n)$ and $d=d(n)$, both of which tend to infinity, for which the sequence ${\cal P}_l^d$ has threshold on the order of $N$.

Min. DegreeFinally one might consider the behavior of graphs of high minimum degree. Define ${\cal G}(n,\delta)$ to be the set of all connected graphs on $n$ vertices having minimum degree at least $\delta = \delta(n)$. Let ${\cal G}_{\delta}=(G_1,\ldots,G_n,\ldots)$ denote any sequence of graphs with each $G_n \in {\cal G}(n,\delta)$. In [CzHu1] is proven the following.Min. Degree Thresholds. For every function $N^{1/2} \ll \delta = \delta(N) \leq N-1$, $\tau({\cal G}_{\delta}) \subset O(N^{3/2}/\delta)$. In particular, if in addition $\delta \in \Omega(N)$ then $\tau({\cal G}_{\delta})=\Theta(N^{1/2})$.