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Articles 1 - 9 of 9
Full-Text Articles in Physical Sciences and Mathematics
Combinatorially Orthogonal Paths, Sean A. Bailey, David E. Brown, Leroy Beaseley
Combinatorially Orthogonal Paths, Sean A. Bailey, David E. Brown, Leroy Beaseley
Communications on Number Theory and Combinatorial Theory
Vectors x=(x1,x2,...,xn)T and y=(y1,y2,...,yn)T are combinatorially orthogonal if |{i:xiyi≠0}|≠1. An undirected graph G=(V,E) is a combinatorially orthogonal graph if there exists f:V→ℝn such that for any u,v∈V, uv∉E iff f(u) and f(v) are combinatorially orthogonal. We will show that every graph has a combinatorially orthogonal representation. We will show …
A Pebbling Game On Powers Of Paths, Garth Isaak, Matthew Prudente, Joseph M. Marcinik Iii
A Pebbling Game On Powers Of Paths, Garth Isaak, Matthew Prudente, Joseph M. Marcinik Iii
Communications on Number Theory and Combinatorial Theory
Two Player Graph Pebbling is an extension of graph pebbling. Players Mover and Defender use pebbling moves, the act of removing two pebbles from one vertex and placing one pebble on an adjacent vertex, to win. If a specified vertex has a pebble on it, then Mover wins. If a specified vertex is pebble-free and there are no more valid pebbling moves, then Defender wins. The Two-Player Pebbling Number of a graph G, η(G), is the minimum m such that for every arrangement of m pebbles and for any specified vertex, Mover can win. We specify the …
Meertens Number And Its Variations, Chai Wah Wu
Meertens Number And Its Variations, Chai Wah Wu
Communications on Number Theory and Combinatorial Theory
In 1998, Bird introduced Meertens numbers as numbers that are invariant under a map similar to the Gödel encoding. In base 10, the only known Meertens number is 81312000. We look at some properties of Meertens numbers and consider variations of this concept. In particular, we consider variations of Meertens numbers where there is a finite time algorithm to decide whether such numbers exist, exhibit infinite families of these variations and provide bounds on parameters needed for their existence.
On Two-Player Pebbling, Garth Isaak, Matthew Prudente, Andrea Potylycki, William Fagley, Joseph Marcinik
On Two-Player Pebbling, Garth Isaak, Matthew Prudente, Andrea Potylycki, William Fagley, Joseph Marcinik
Communications on Number Theory and Combinatorial Theory
Graph pebbling can be extended to a two-player game on a graph G, called Two-Player Graph Pebbling, with players Mover and Defender. The players each use pebbling moves, the act of removing two pebbles from one vertex and placing one of the pebbles on an adjacent vertex, to win. Mover wins if they can place a pebble on a specified vertex. Defender wins if the specified vertex is pebble-free and there are no more pebbling moves on the vertices of G. The Two-Player Pebbling Number of a graph G, η(G), is the minimum m such that for every arrangement …
Some Np-Complete Edge Packing And Partitioning Problems In Planar Graphs, Jed Yang
Some Np-Complete Edge Packing And Partitioning Problems In Planar Graphs, Jed Yang
Communications on Number Theory and Combinatorial Theory
Graph packing and partitioning problems have been studied in many contexts, including from the algorithmic complexity perspective. Consider the packing problem of determining whether a graph contains a spanning tree and a cycle that do not share edges. Bernáth and Király proved that this decision problem is NP-complete and asked if the same result holds when restricting to planar graphs. Similarly, they showed that the packing problem with a spanning tree and a path between two distinguished vertices is NP-complete. They also established the NP-completeness of the partitioning problem of determining whether the edge set of a graph can be …
Generating B-Nomial Numbers, Ji Young Choi
Generating B-Nomial Numbers, Ji Young Choi
Communications on Number Theory and Combinatorial Theory
This paper presents three new ways to generate each type of b-nomial numbers: We develop ordinary generating functions, we find a whole new set of recurrence relations, and we identify each b-nomial number as a single binomial coefficient or as an alternating sum of products of two binomial coefficients.
Skolem Number Of Subgraphs On The Triangular Lattice, Braxton Carrigan, Garrett Green
Skolem Number Of Subgraphs On The Triangular Lattice, Braxton Carrigan, Garrett Green
Communications on Number Theory and Combinatorial Theory
A Skolem sequence can be thought of as a labelled path where any two vertices with the same label are that distance apart. This concept has naturally been generalized to graph labelling. This brings rise to the question; “what is the smallest set of consecutive positive integers we can use to properly Skolem label a graph?” This is known as the Skolem number of the graph. In this paper we give the Skolem number for three natural vertex induced subgraphs of the triangular lattice graph.
Determining Biases In The Card-Chameleon Cryptosystem, Isaac Reiter, Eric Landquist
Determining Biases In The Card-Chameleon Cryptosystem, Isaac Reiter, Eric Landquist
Communications on Number Theory and Combinatorial Theory
Throughout history, spies, soldiers, and others have relied on so-called {\em hand ciphers} to send encrypted messages. Since the creation of Pontifex (also known as Solitaire) by Bruce Schneier in 1999, a number of hand ciphers utilizing a standard deck of playing cards have emerged. Since there are $52! \approx 2^{225.58}$ possible ways to order a deck of cards, there are over 225 bits of entropy in a well-shuffled deck of cards. Theoretically, this can provide enough security to rival modern computer-based cryptosystems. In this paper, we describe and analyze one such playing card cipher, Card-Chameleon, created by Matthew McKague. …
Super Sudoku Squares, Dean Hoffman
Super Sudoku Squares, Dean Hoffman
Communications on Number Theory and Combinatorial Theory
We refer to a completed Sudoku puzzle as a Sudoku Square of order 3. We define a more restricted, but natural, extension --- a Super Sudoku Square of order n, and determine for which orders n one exists.