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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. [ABSTRACT FROM AUTHOR]

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 byt a vertex. Deciding if the pebbling number is at most k is $$\Pi _2^\mathsf{P}$$ -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, $$4\mathrm{th}$$ weak Bruhat, and Lemke squared, 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. In doing so we partly answer 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. [ABSTRACT FROM AUTHOR]

A Gray code is a listing structure for a set of combinatorial objects such that some consistent (usually minimal) change property is maintained throughout adjacent elements in the list. While Gray codes for m-ary strings have been considered in the past, we provide a new, simple Gray code for fixed-weight m-ary strings. In addition, we consider a relatively new type of Gray code known as overlap cycles and prove basic existence results concerning overlap cycles for fixed-weight and weight-range m-ary words. [ABSTRACT FROM AUTHOR]

A number of applications of Steiner triple systems (e.g. disk erasure codes) exist that require a special ordering of its blocks. Universal cycles, introduced by Chung, Diaconis, and Graham in 1992, and Gray codes are examples of listing elements of a combinatorial family in a specific manner, and Godbole invented the following generalization of these in 2010. 1-overlap cycles require a set of strings to be ordered so that the last letter of one string is the first letter of the next. In this paper, we prove the existence of 1-overlap cycles for automorphism free Steiner triple systems of each possible order. Since Steiner triple systems have the property that each block can be represented uniquely by a pair of points, these 1-overlap cycles can be compressed by omitting non-overlap points to produce rank two universal cycles on such designs, expanding on the results of Dewar. [ABSTRACT FROM AUTHOR]

Graph pebbling is a network optimization model for transporting discrete resources that are consumed in transit: the movement of 2 pebbles across an edge consumes one of the pebbles. The pebbling number of a graph is the fewest number of pebbles t so that, from any initial configuration of t pebbles on its vertices, one can place a pebble on any given target vertex via such pebbling steps. It is known that deciding whether a given configuration on a particular graph can reach a specified target is NP-complete, even for diameter 2 graphs, and that deciding whether the pebbling number has a prescribed upper bound is Πp2-complete. On the other hand, for many families of graphs there are formulas or polynomial algorithms for computing pebbling numbers; for example, complete graphs, products of paths (including cubes), trees, cycles, diameter 2 graphs, and more. Moreover, graphs having minimum pebbling number are called Class 0, and many authors have studied which graphs are Class 0 and what graph properties guarantee it, with no characterization in sight. In this paper we investigate an important family of diameter 3 chordal graphs called split graphs; graphs whose vertex set can be partitioned into a clique and an independent set. We provide a formula for the pebbling number of a split graph, along with an algorithm for calculating it that runs in O(nβ) time, where β = 2ω/(ω +1) #8773; 1.41 and ω ≅ 2.376 is the exponent of matrix multiplication. Furthermore we determine that all split graphs with minimum degree at least 3 are Class 0. [ABSTRACT FROM AUTHOR]

Steiner quadruple systems are set systems in which every triple is contained in a unique quadruple. It is well known that Steiner quadruple systems of order v, or SQS( v), exist if and only if [ABSTRACT FROM AUTHOR]

Universal cycles are generalizations of de Bruijn cycles and Gray codes that were introduced originally by Chung, Diaconis, and Graham in 1992. They have been developed by many authors since, for various combinatorial objects such as strings, subsets, permutations, partitions, vector spaces, and designs. One generalization of universal cycles, which require almost complete overlap of consecutive words, is s-overlap cycles, which relax such a constraint. In this paper we study weak orders, which are relations that are transitive and complete. We prove the existence of universal and s-overlap cycles for weak orders, as well as for fixed height and/or weight weak orders, and apply the results to cycles for ordered partitions. [ABSTRACT FROM AUTHOR]

Graph pebbling is a game played on a connected graph G. A player purchases pebbles at a dollar a piece and hands them to an adversary who distributes them among the vertices of G (called a configuration) and chooses a target vertex r. The player may make a pebbling move by taking two pebbles off of one vertex and moving one of them to a neighboring vertex. The player wins the game if he can move k pebbles to r. The value of the game ( G, k), called the k-pebbling number of G and denoted ππk( G), is the minimum cost to the player to guarantee a win. That is, it is the smallest positive integer m of pebbles so that, from every configuration of size m, one can move k pebbles to any target. In this paper, we use the block structure of graphs to investigate pebbling numbers, and we present the exact pebbling number of the graphs whose blocks are complete. We also provide an upper bound for the k-pebbling number of diameter-two graphs, which can be the basis for further investigation into the pebbling numbers of graphs with blocks that have diameter at most two. [ABSTRACT FROM AUTHOR]

Let S be a cyclic n–ary sequence. We say that S is a 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 (Sn) of (n, k)–Ucycle packings for fixed k is a near–Ucycle if the length of Sn is (1 – o(1)) (n k) . In this paper we prove that near–(n, k)–Ucycles exist for all k. [ABSTRACT FROM AUTHOR]

The pebbling number of a graph is the smallest number t such that from any initial configuration of t pebbles one can move a pebble to any prescribed vertex by a sequence of pebbling steps. It is known that graphs whose connectivity is high compared to their diameter have a pebbling number as small as possible. We will use the above result to prove two related theorems. First, answering a question of the second author, we show that there exist graphs of arbitrarily high constant girth and least possible pebbling number. In the second application, we prove that the product of two graphs of high minimum degree has a pebbling number equal to the number of vertices of the product. This shows that Graham's product conjecture is true in the case of high minimum degree graphs. In addition, we consider a probabilistic variant of the pebbling problem and establish a pebbling threshold result for products of paths. The last result shows that the sequence of paths satisfies the probabilistic analogue of Graham's product conjecture. [ABSTRACT FROM AUTHOR]

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)=Ω(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. [ABSTRACT FROM AUTHOR]

A partially ordered set P is called a circle containment order provided one can assign to each x∈ P a circle Cso that $$x \leqslant y \Leftrightarrow C_x \subseteq C_y $$ . We show that the infinite three-dimensional poset N is not a circle containment order and note that it is still unknown whether or not [ n] is such an order for arbitrarily large n. [ABSTRACT FROM AUTHOR]