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Graph pebbling is a game played on graphs with pebbles on their vertices. A pebbling move removes two pebbles from one vertex and places one pebble on an adjacent vertex. The pebbling number $\pi(G)$ is the smallest $t$ so that from any initial configuration of $t$ pebbles it is possible, after a sequence of pebbling moves, to place a pebble on any given target vertex. In this paper, we provide the first results on the pebbling numbers of snarks.

Here we merge the two fields of Cops and Robbers and Graph Pebbling to introduce the new topic of Cops and Robbers Pebbling. Both paradigms can be described by moving tokens (the cops) along the edges of a graph to capture a special token (the robber). In Cops and Robbers, all tokens move freely, whereas, in Graph Pebbling, some of the chasing tokens disappear with movement while the robber is stationary. In Cops and Robbers Pebbling, some of the chasing tokens (cops) disappear with movement, while the robber moves freely. We define the cop pebbling number of a graph to be the minimum number of cops necessary to capture the robber in this context, and present upper and lower bounds and exact values, some involving various domination parameters, for an array of graph classes. We also offer several interesting problems and conjectures.

Given a graph $G$, let $\mu(G)$ denote the size of the smallest maximal independent set in $G$. A family of subsets is called a star if some element is in every set of the family. A split vertex has degree at least 3. Holroyd and Talbot conjectured the following Erd\H{o}s-Ko-Rado type statement about intersecting families of independent sets in graphs: if $1\le r\le \mu(G)/2$ then there is an intersecting family of independent $r$-sets of maximum size that is a star. In this paper we prove similar statements for sparse graphs on $n$ vertices: roughly, for graphs of bounded average degree with $r\le O(n^{1/3})$, for graphs of bounded degree with $r\le O(n^{1/2})$, and for trees having a bounded number of split vertices with $r\le O(n^{1/2})$., Comment: Correction of typos and inclusion of additional history

A new graph invariant called the secure vertex cover pebbling number, which is a combination of two graph invariants, namely secure vertex cover and cover pebbling number, is introduced in this paper. The secure vertex cover pebbling number of a graph G is the minimum number m so that every distribution of m pebbles can reach some secure vertex cover of G by a sequence of pebbling moves. In this paper, the complexity of the secure vertex cover problem and secure vertex cover pebbling problem are discussed. Also, we obtain some basic results and the secure vertex cover pebbling number for complete r- partite graphs, paths, Friendship graphs, and wheel graphs., Comment: 19 pages, 3 figures

Graph pebbling is a network optimization model for satisfying vertex demands with vertex supplies (called pebbles), with partial loss of pebbles in transit. The pebbling number of a demand in a graph is the smallest number for which every placement of that many supply pebbles satisfies the demand. The Target Conjecture (Herscovici-Hester-Hurlbert, 2009) posits that the largest pebbling number of a demand of fixed size $t$ occurs when the demand is entirely stacked on one vertex. This truth of this conjecture could be useful for attacking many open problems in graph pebbling, including the famous conjecture of Graham (1989) involving graph products. It has been verified for complete graphs, cycles, cubes, and trees. In this paper we prove the conjecture for 2-paths and Kneser graphs over pairs., Comment: 16 pages, 4 figures. Replaces "The Target Pebbling Conjecture" (2011.10623.v2), with improved exposition

Graph pebbling models the transportation of consumable resources. As two pebbles move across an edge, one reaches its destination while the other is consumed. The $t$-pebbling number is the smallest integer $m$ so that any initially distributed supply of $m$ pebbles can place $t$ pebbles on any target vertex via pebbling moves. The 1-pebbling number of diameter two graphs is well-studied. Here we investigate the $t$-pebbling number of diameter two graphs under the lense of connectivity., 6 pages, 1 figure

A family of sets is intersecting if any two sets in the family intersect. Given a graph $G$ and an integer $r\geq 1$, let $\mathcal{I}^{(r)}(G)$ denote the family of independent sets of size $r$ of $G$. For a vertex $v$ of $G$, the family of independent sets of size $r$ that contain $v$ is called an $r$-star. Then $G$ is said to be $r$-EKR if no intersecting subfamily of $ \mathcal{I}^{(r)}(G)$ is bigger than the largest $r$-star. Let $n$ be a positive integer, and let $G$ consist of the disjoint union of $n$ paths each of length 2. We prove that if $1 \leq r \leq n/2$, then $G$ is $r$-EKR. This affirms a longstanding conjecture of Holroyd and Talbot for this class of graphs and can be seen as an analogue of a well-known theorem on signed sets, proved using different methods, by Deza and Frankl and by Bollob��s and Leader. Our main approach is a novel probabilistic extension of Katona's elegant cycle method, which might be of independent interest., 12 pages, 1 figure. To appear in Discrete Mathematics

Let ${\mathscr L}=(X,\preceq)$ be a lattice. For ${\cal P}\subseteq X$ we say that ${\cal P}$ is $t$-{\it intersecting} if ${\sf rank}(x\wedge y)\ge t$ for all $x,y\in{\cal P}$. The seminal theorem of Erd��s, Ko and Rado describes the maximum intersecting ${\cal P}$ in the lattice of subsets of a finite set with the additional condition that ${\cal P}$ is contained within a level of the lattice. The Erd��s-Ko-Rado theorem has been extensively studied and generalized to other objects and lattices. In this paper, we focus on intersecting families of permutations as defined with respect to the weak Bruhat lattice. In this setting, we prove analogs of certain extremal results on intersecting set systems. In particular we give a characterization of the maximum intersecting families of permutations in the Bruhat lattice. We also characterize the maximum intersecting families of permutations within the $r^{\textrm{th}}$ level of the Bruhat lattice of permutations of size $n$, provided that $n$ is large relative to $r$.

Pebbling on graphs is a two-player game which involves repeatedly moving a pebble from one vertex to another by removing another pebble from the first vertex. The pebbling number $\pi(G)$ is the least number of pebbles required so that, regardless of the initial configuration of pebbles, a pebble can reach any vertex. Graham conjectured that the pebbling number for the cartesian product, $G \hspace{1mm}\square\hspace{1mm} H$, is bounded above by $\pi(G) \pi(H)$. We show that $\pi(G\hspace{1mm}\square\hspace{1mm} H) \le 2\pi(G) \pi(H)$ and, more sharply, that $\pi(G \hspace{1mm}\square\hspace{1mm} H) \le (\pi(G)+|G|) \pi(H)$. Furthermore, we provide similar results for other graph products and graph operations., Comment: 20 pages, 5 figures

A starting point in the investigation of intersecting systems of subsets of a finite set is the elementary observation that the size of a family of pairwise intersecting subsets of a finite set [n]={1,...,n}, denoted by 2^{[n]}, is at most 2^{n-1}, with one of the extremal structures being the family comprised of all subsets of [n] containing a fixed element, called as a star. A longstanding conjecture of Chv��tal aims to generalize this simple observation for all downsets of 2^{[n]}. In this note, we prove this conjecture for all downsets where every subset contains at most 3 elements., 10 pages, 0 figures

A set system F is intersecting if any pair of sets in F have a nonempty intersection. A fundamental theorem of Erd��s, Ko and Rado states that if F is an intersecting family of r-subsets of [n]={1,...,n}, and n>= 2r, then the cardinality of F is at most the cardinality of the family of all r-subsets of [n] containing a fixed element. Furthermore, when n>2r, equality holds if and only if F is the family of all r-subsets of [n] containing a fixed element. This characterization was proved as part of a stronger result by Hilton and Milner. In this note, we provide new injective proofs of the Erd��s--Ko--Rado and the Hilton--Milner theorems., 9 pages. Section 4 is a new addition, containing a new proof of the Hilton--Milner theorem. The presentation of the proof of the EKR theorem in Section 3 is also slightly changed. Additional relevant references have been added

The goal of this paper is to solve Problem 481 from the list of research problems in the special issue of Discrete Mathematics dedicated to the Banff International Research Station workshop on "Generalizations of de Bruijn Cycles and Gray Codes" in 2004. Overlap cycles are generalizations of de Bruijn cycles and Gray codes that were introduced originally in 2010 by Godbole et al. In this paper we prove that s-overlap cycles for k-permutations of [n] exist for all k, Comment: 6 pages; Corrected fatal flaw in proof of Lemma 3

Steiner quadruple systems are set systems in which every triple is contained in a unique quadruple. It is will known that Steiner quadruple systems of order v, or SQS(v), exist if and only if v = 2, 4 mod 6. Universal cycles, introduced by Chung, Diaconis, and Graham in 1992, are a type of cyclic Gray code. Overlap cycles are generalizations of universal cycles that were introduced in 2010 by Godbole. Using Hanani's SQS constructions, we show that for every v = 2, 4 mod 6 with v > 4 there exists an SQS(v) that admits a 1-overlap cycle., 24 pages

Graph pebbling is a network model for studying whether or not a given supply of discrete pebbles can satisfy a given demand via pebbling moves. A pebbling move across an edge of a graph takes two pebbles from one endpoint and places one pebble at the other endpoint; the other pebble is lost in transit as a toll. It has been shown that deciding whether a supply can meet a demand on a graph is NP-complete. The pebbling number of a graph is the smallest t such that every supply of t pebbles can satisfy every demand of one pebble. Deciding if the pebbling number is at most k is \Pi_2^P-complete. In this paper we develop a tool, called the Weight Function Lemma, for computing upper bounds and sometimes exact values for pebbling numbers with the assistance of linear optimization. With this tool we are able to calculate the pebbling numbers of much larger graphs than in previous algorithms, and much more quickly as well. We also obtain results for many families of graphs, in many cases by hand, with much simpler and remarkably shorter proofs than given in previously existing arguments (certificates typically of size at most the number of vertices times the maximum degree), especially for highly symmetric graphs. Here we apply the Weight Function Lemma to several specific graphs, including the Petersen, Lemke, 4th weak Bruhat, Lemke squared, and two random graphs, as well as to a number of infinite families of graphs, such as trees, cycles, graph powers of cycles, cubes, and some generalized Petersen and Coxeter graphs. This partly answers a question of Pachter, et al., by computing the pebbling exponent of cycles to within an asymptotically small range. It is conceivable that this method yields an approximation algorithm for graph pebbling.

We prove a generalization of Graham's Conjecture for optimal pebbling with arbitrary sets of target distributions. We provide bounds on optimal pebbling numbers of products of complete graphs and explicitly find optimal $t$-pebbling numbers for specific such products. We obtain bounds on optimal pebbling numbers of powers of the cycle $C_5$. Finally, we present explicit distributions which provide asymptotic bounds on optimal pebbling numbers of hypercubes., Comment: 28 pages, 1 figure

One of the more recent generalizations of the Erd��s-Ko-Rado theorem, formulated by Holroyd, Spencer and Talbot, defines the Erd��s-Ko-Rado property for graphs in the following manner: for a graph G and a positive integer r, G is said to be r-EKR if no intersecting subfamily of the family of all independent vertex sets of size r is larger than the largest star, where a star centered at a vertex v is the family of all independent sets of size $r$ containing v. In this paper, we prove that if G is a disjoint union of chordal graphs, including at least one singleton, then G is r-EKR if $r\leq mu(G)/2$, where mu(G) is the minimum size of a maximal independent set. We will also prove Erd��s-Ko-Rado results for chains of complete graphs, which are a class of chordal graphs obtained by blowing up edges of a path into complete graphs. We also consider similar problems for ladder graphs and trees, and prove preliminary results for these graphs., 30 pages, 5 figures. This is the second version. Conjecture 4.1 from the previous version has been disproved, and the relevant section has been accordingly modified

We investigate generalizations of pebbling numbers and of Graham's pebbling conjecture that pi(GxH), 18 pages

Let S be a cyclic n-ary sequence. We say that S is a {\it universal cycle} ((n,k)-Ucycle) for k-subsets of [n] if every such subset appears exactly once contiguously in S, and is a Ucycle packing if every such subset appears at most once. Few examples of Ucycles are known to exist, so the relaxation to packings merits investigation. A family {S_n} of (n,k)-Ucycle packings for fixed k is a near-Ucycle if the length of S_n is $(1-o(1))\binom{n}{k}$. In this paper we prove that near-(n,k)-Ucycles exist for all k., 14 pages, 3 figures

Suppose 2n voters vote sequentially for one of two candidates. For how many such sequences does one candidate have strictly more votes than the other at each stage of the voting? The answer is \binom{2n}{n} and, while easy enough to prove using generating functions, for example, only two combinatorial proofs exist, due to Kleitman and Gessel. In this paper we present two new (far simpler) bijective proofs.

The subject of graph pebbling has seen dramatic growth recently, both in the number of publications and in the breadth of variations and applications. Here we update the reader on the many developments that have occurred since the original Survey of Graph Pebbling in 1999., Comment: 24 pages, 1 figure

Given a graph G and a configuration C of pebbles on the vertices of G, a pebbling step removes two pebbles from one vertex and places one pebble on an adjacent vertex. The cover pebbling number g=g(G) is the minimum number so that every configuration of g pebbles has the property that, after some sequence of pebbling steps, every vertex has a pebble on it. We prove that the cover pebbling number of the d-dimensional hypercube Q^d equals 3^d., 11 pages

We say that a graph G is Class 0 if its pebbling number is exactly equal to its number of vertices. For a positive integer d, let k(d) denote the least positive integer so that every graph G with diameter at most d and connectivity at least k(d) is Class 0. The existence of the function k was conjectured by Clarke, Hochberg and Hurlbert, who showed that if the function k exists, then it must satisfy k(d)=\Omega(2^d/d). In this note, we show that k exists and satisfies k(d)=O(2^{2d}). We also apply this result to improve the upper bound on the random graph threshold of the Class 0 property., Comment: 12 pages

We survey results on the pebbling numbers of graphs as well as their historical connection with a number-theoretic question of Erd\H os and Lemke. We also present new results on two probabilistic pebbling considerations, first the random graph threshold for the property that the pebbling number of a graph equals its number of vertices, and second the pebbling threshold function for various natural graph sequences. Finally, we relate the question of the existence of pebbling thresholds to a strengthening of the normal property of posets, and show that the multiset lattice is not supernormal., 24 pages

A configuration of pebbles on the vertices of a graph is solvable if one can place a pebble on any given root vertex via a sequence of pebbling steps. The pebbling number of a graph G is the minimum number pi(G) so that every configuration of pi(G) pebbles is solvable. A graph is Class 0 if its pebbling number equals its number of vertices. A function is a pebbling threshold for a sequence of graphs if a randomly chosen configuration of asymptotically more pebbles is almost surely solvable, while one of asymptotically fewer pebbles is almost surely not. Here we prove that graphs on n>=9 vertices having minimum degree at least floor(n/2) are Class 0, as are bipartite graphs with m>=336 vertices in each part having minimum degree at least floor(m/2)+1. Both bounds are best possible. In addition, we prove that the pebbling threshold of graphs with minimum degree d, with sqrt{n} << d, is O(n^{3/2}/d), which is tight when d is proportional to n., 10 pages

A configuration of pebbles on the vertices of a graph is solvable if one can place a pebble on any given root vertex via a sequence of pebbling steps. A function is a pebbling threshold for a sequence of graphs if a randomly chosen configuration of asymptotically more pebbles is almost surely solvable, while one of asymptotically fewer pebbles is almost surely not. In this note we show that the spectrum of pebbling thresholds for graph sequences spans the entire range from n^{1/2} to n. This answers a question of Czygrinow, Eaton, Hurlbert and Kayll. What the spectrum looks like above n remains unknown., Comment: 8 pages, preliminary version

A sequence of elements of a finite group G is called a zero-sum sequence if it sums to the identity of G. The study of zero-sum sequences has a long history with many important applications in number theory and group theory. In 1989 Kleitman and Lemke, and independently Chung, proved a strengthening of a number theoretic conjecture of Erdos and Lemke. Kleitman and Lemke then made more general conjectures for finite groups, strengthening the requirements of zero-sum sequences. In this paper we prove their conjecture in the case of abelian groups. Namely, we use graph pebbling to prove that for every sequence (g_k)_{k=1}^{|G|} of |G| elements of a finite abelian group G there is a nonempty subsequence (g_k)_{k in K} such that sum_{k in K}g_k=0_G and sum_{k in K}1/|g_k|\le 1, where |g| is the order of the element g in G., Comment: 18 pages