John GoldwasserWest Virginia University | WVU · Department of Mathematics

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Publications (59)
Article
Let be the hypercube of dimension and let and be subsets of the vertex set , called configurations in . We say that is an exact copy of if there is an automorphism of which sends onto . Let be an integer, let be a configuration in and let be a configuration in . We let be the maximum, over all configurations in , of the fraction of sub‐‐cubes of in...
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Let $Q_d$ be the hypercube of dimension $d$ and let $H$ and $K$ be subsets of the vertex set $V(Q_d)$, called configurations in $Q_d$. We say that $K$ is an \emph{exact copy} of $H$ if there is an automorphism of $Q_d$ which sends $H$ onto $K$. Let $n\geq d$ be an integer, let $H$ be a configuration in $Q_d$ and let $S$ be a configuration in $Q_n$....
Article
If G is a graph and H is a set of subgraphs of G, we say that an edge-coloring of G is H-polychromatic if every graph from H gets all colors present in G on its edges. The H-polychromatic number of G, denoted by polyH(G), is the largest number of colors in an H-polychromatic coloring. In this paper we determine polyH(G) exactly when G is a complete...
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If S is a subset of an abelian group G, the polychromatic number of S in G is the largest integer k so that there is a k-coloring of the elements of G such that every translate of S in G gets all k colors. We determine the polychromatic number of all sets of size 2 or 3 in the group of integers mod\documentclass[12pt]{minimal} \usepackage{amsmath}...
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Let H and K be subsets of the vertex set V(Qd) of the d-cube Qd (we call H and K configurations in Qd). We say K is an exact copy of H if there is an automorphism of Qd which sends H to K. If d is a positive integer and H is a configuration in Qd, we define λ(H,d) to be the limit as n goes to infinity of the maximum fraction, over all subsets S of...
Preprint
If $G$ is a graph and $\mathcal{H}$ is a set of subgraphs of $G$, we say that an edge-coloring of $G$ is $\mathcal{H}$-polychromatic if every graph from $\mathcal{H}$ gets all colors present in $G$ on its edges. The $\mathcal{H}$-polychromatic number of $G$, denoted $\operatorname{poly}_\mathcal{H} (G)$, is the largest number of colors in an $\math... Preprint Let$H$and$K$be subsets of the vertex set$V(Q_d)$of the$d$-cube$Q_d$(we call$H$and$K$configurations in$Q_d$). We say$K$is an \emph{exact copy} of$H$if there is an automorphism of$Q_d$which sends$H$to$K$. If$d$is a positive integer and$H$is a configuration in$Q_d$, we define$\pi(H,d)$to be the limit as$n$goes to infini... Preprint Full-text available If$S$is a subset of an abelian group$G$, the polychromatic number of$S$in$G$is the largest integer$k$so that there is a$k-$coloring of the elements of$G$such that every translate of$S$in$G$gets all$k$colors. We determine the polychromatic number of all sets of size 2 or 3 in the group of integers mod n. Article Two sets are weakly incomparable if neither properly contains the other; they are strongly incomparable if they are unequal and neither contains the other. Two families A and B of sets are weakly (or strongly) incomparable if no set in one of A and B is weakly (or strongly) comparable to a set in the other. A family A of sets is uncomplemented if A... Article The game LIGHTS OUT! is played on a 5×5 square grid of buttons; each button may be on or off. Pressing a button changes the on/off state of the light of the button pressed and of all its vertical and horizontal neighbors. Given an initial configuration of buttons that are on, the object of the game is to turn all the lights out. The game can be gen... Article Full-text available We show that for any set$S\subseteq \mathbb{Z}$,$|S|=4$there exists a 3-coloring of$\mathbb{Z}$in which every translate of$S$receives all three colors. This implies that$S$has a codensity of at most$1/3\$, proving a conjecture of Newman [D. J. Newman, Complements of finite sets of integers, Michigan Math. J. 14 (1967) 481--486]. We also co...
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If G is a graph and H is a set of subgraphs of G, then an edge-coloring of G is called H-polychromatic if every graph from H gets all colors present in G on its edges. The H-polychromatic number of G, denoted poly_H(G), is the largest number of colors in an H-polychromatic coloring. In this paper, poly_H(G) is determined exactly when G is a complet...
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Given a subgraph G of the hypercube Q_n, a coloring of the edges of Q_n such that every embedding of G contains an edge of every color is called a G-polychromatic coloring. The maximum number of colors with which it is possible to G-polychromatically color the edges of any hypercube is called the polychromatic number of G. To determine polychromati...
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For a fixed M x N integer lattice L(M,N), we consider the maximum size of a subset A of L(M,N) which contains no squares of prescribed side lengths k(1),...,k(t). We denote this size by ex(L(M,N), {k(1),...,k(t)}), and when t = 1, we abbreviate this parameter to ex(L(M,N), k), where k = k(1). Our first result gives an exact formula for ex(L(M,N), k...
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We generalize a theorem of M. Hall Jr., that an Latin rectangle on n symbols can be extended to an Latin square on the same n symbols. Let p, n, be positive integers such that and . Call an matrix on n symbols an -latinized rectangle if no symbol occurs more than once in any row or column, and if the symbol occurs at most times altogether . We give...
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Eternal domination of a graph requires the vertices of the graph to be protected, against infinitely long sequences of attacks, by guards located at vertices, with the requirement that the configuration of guards induces a dominating set at all times. We present results and conjectures for grid graphs for some variations on this problem.
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We prove that a maximum subset of (1,2,…, n) containing no solutions to x+y=3z has n/2 elements if n≠4, thus settling a conjecture of Erdős. For n≥23 the set of all odd integers less than or equal to n is the unique maximum such subset.
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If H is a 3-graph, then ex(n;H) denotes the maximum number of edges in a 3-graph on n vertices containing no sub-3-graph isomorphic to H. Let S(n) denote the 3-graph on n vertices obtained by partitioning the vertex set into parts of sizes ⌈n 2⌉ and ⌊n 2⌋ and taking as edges all triples that intersect both parts. Let s(n) denote the number of edges...
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If we 2-color the vertices of a large hypercube what monochromatic substructures are we guaranteed to find? Call a set S of vertices from Q_d, the d-dimensional hypercube, Ramsey if any 2-coloring of the vertices of Q_n, for n sufficiently large, contains a monochromatic copy of S. Ramsey's theorem tells us that for any r \geq 1 every 2-coloring of...
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Consider a graph each of whose vertices is either in the ON state or in the OFF state and call the resulting ordered bipartition into ON vertices and OFF vertices a configuration of the graph. A regular move at a vertex changes the states of the neighbors of that vertex and hence sends the current configuration to another one. A valid move is a reg...
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In 1974 Cruse gave necessary and sufficient conditions for an r × s partial latin square P on symbols σ1,σ2,…,σt, which may have some unfilled cells, to be completable to an n × n latin square on symbols σ1,σ2,…,σn, subject to the condition that the unfilled cells of P must be filled with symbols chosen from {σt + 1,σt + 2,…,σn}. These conditions c...
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Hall's condition is a well-known necessary condition for the existence of a proper coloring of a graph from prescribed lists. Completing a partial latin square is a very special kind of graph list-coloring problem. Cropper's question was: is Hall's condition sufficient for the existence of a completion of a partial latin square? The folk belief tha...
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Let G be a graph in which each vertex can be in one of two states: on or off. In the σ-game, when you “push” a vertex v you change the state of all of its neighbors, while in the σ+-game you change the state of v as well. Given a starting configuration of on vertices, the object of both games is to reduce it, by a sequence of pushes, to the smalles...
Article
Each vertex in a simple graph is in one of two states: “on” or “off”. The set of all on vertices is called a configuration. In the σ-game, “pushing” a vertex v changes the state of all vertices in the open neighborhood of v, while in the σ+-game pushing v changes the state of all vertices in its closed neighborhood. The reachability question for th...
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The eternal domination number of a graph is the number of guards needed at vertices of the graph to defend the graph against any sequence of attacks at vertices. We consider the model in which at most one guard can move per attack and a guard can move across at most one edge to defend an attack. We prove that there are graphs G for which γ∞(G)⩾α(G)...
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Abstract A subset D of the vertex set V of a graph is called an open oddd dominating set if each vertex in V is adjacent to an odd number,of vertices in D (adjacency is irre∞exive). In this paper we solve the existence and enumeration problems for odd open dominating sets (and analogously deflned even open dominating sets) in the m £ n grid graph a...
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A total perfect code in a simple, undirected graph G=(V,E) is a set D⊆V such that |N(v)∩D|=1 for all v∈V. In this paper, we characterize which grid graphs have total perfect codes.
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Let X={1,2,…,n}, 2kn, and let X(k)denote the set of all subsets of Xof size k. A set system F⊆X(k)is intersecting if no two of its elements are disjoint. The minimum complementary degree c(F)of Fis the minimum over all i∈Xof the number of sets in Fnot containing i. A set system F⊆X(k)is complementary degree condition maximal (CDCM) if H⊆X(k)interse...
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Abstract A positive even closed (open) dominating set of a graph G is a subset D of the vertices such that each vertex in G has a positive even number,of vertices of D in its closed (open) neighborhood. In this paper we flnd all positive even open dominating sets in the m£n grid graph Gm;n and, with the help of a computer, flnd all m;n such that Gm...
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If L is a list assignment of colors to the vertices of a graph G of chromatic number χ(G), a certain condition on L and G, known as Hall's condition, which is obviously necessary for G to have an L-coloring, is known to be sufficient if and only if each block of G is a clique. We show that if the set of colors from which the lists are drawn has siz...
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A non-empty set of vertices is called an even dominating set if each vertex in the graph is adjacent to an even number of vertices in the set (adjacency is reexive). In this paper, the Fibonacci polynomials are studied over GF (2) with particular emphasis on their divisibility properties and their relation to the existence of even dominating sets i...
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Let G = (V, E) be a simple, undirected graph. A set of verticesD is called an odd dominating set if |N [v] ∩ D| ≡ 1(mod 2) for every vertex v ∈ V (G). The minimum cardinality of an odd dominating setis called the odd domination number of G, denoted by γ_1(G). In this paper, several algorithmic and structural results are presented on this parameter...
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Given an n by m rectangular array of switches, some in the on position and some in the off position, we wish to achieve the all-off configuration using an activate operation. The activate operation consists of activating a particular switch which causes it and its rectilinearly adjacent neighbors to change state. By investigating the null-space of...
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Complexity results and algorithms are given for the problem of maximizing the number of off vertices (switches) in graphs and m Theta n rectangular grids. When a switch is toggled, it and its neighbors change state. It is shown that the problem is NP-complete in graphs and a simple approximation algorithm is given as well as a nonapproximability re...
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Algebraic conditions and algorithmic procedures are given to determine whether an m Theta n rectangular configuration of switches can be transformed so that all switches are in the off position, regardless of initial configuration. However, when any switch is toggled, it and its rectilinearly adjacent neighbors change state. Using linear algebra, a...
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Philip Hall's famous theorem on systems of distinct representatives and its not-so-famous improvement by Halmos and Vaughan (1950) can be regarded as statements about the existence of proper list-colorings or list-multicolorings of complete graphs. The necessary and sufficient condition for a proper “coloring” in these theorems has a rather natural...
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A cubic graph G is uniquely edge-3-colorable if G has precisely one 1-factorization. It is proved in this paper, if a uniquely edge-3-colorable, cubic graph G is cyclically 4-edge-connected, but not cyclically 5-edge-connected, then G must contain a snark as a minor. This is an approach to a conjecture that every triangle free uniquely edge-3-colo...
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Pascal's rhombus is a variation of Pascal's triangle in which values are computed as the sum of four terms, rather than two. It is shown that the limiting ratio of the number of ones to the number of zeros in Pascal's rhombus, taken modulo 2, approaches zero. An asymptotic formula for the number of ones in the rhombus is also shown.
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It was proved by Ellingham (1984) that every permutation graph either contains a subdivision of the Petersen graph or is edge-3-colorable. This theorem is an important partial result of Tutte's Edge-3-Coloring Conjecture and is also very useful in the study of the Cycle Double Cover Conjecture. The main result in this paper is that every permutatio...
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If k is a positive real number, we say that a set S of real numbers is k-sum-free if there do not exist x; y; z in S such that x+ y = kz. For k greater than or equal to 4 we find the essentially unique measurable k-sum-free subset of (0; 1] of maximum size.
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If k is a positive real number, we say that a set S of real numbers is k-sum-free if there do not exist x; y; z in S such that x+ y = kz. For k greater than or equal to 4 we find the essentially unique measurable k-sum-free subset of (0; 1] of maximum size. 1 Introduction We say that a set S of real numbers is sum-free if there do not exist x; y; z...
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We use shortened and punctured codes to give an elementary proof of a combinatorial identity of Brualdi, Pless, and Beissinger from which the MacWilliams identities follow as special cases. We also give a short, mostly combinatorial proof of one form of the MacWilliam identities for binary codes.
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Let A be a square matrix and t a positive integer. We say A is t-triangular if there exist permutation matrices P and Q such that PAQ = B = [bij] has bij = 0 whenever j ⩾ i + t. We ask for which positive integers the following statement is true: If A is any square matrix with nonnegative integral entries such that 0 < per A < (t + 1)!, then A is t-...
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Let G be a cubic graph containing no subdivision of the Petersen graph. If G has a 2-factor F consisting of two circuits C 1 and C 2 such that C 1 is chordless and C 2 has at most one chord, then G is edge-3-colorable. This result generalizes an early result by M. N. Ellingham [Congr. Numerantium 44, 33-40 (1984; Zbl 0558.05040)] and is a partial r...
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If k is a positive real number, we say that a set S of real numbers is k-sum-free if there do not exist x;y;z in S such that x +y = kz .F or k greater than or equal to 4 we flnd the essentially unique measurable k-sum-free subset of (0; 1) of maximum size.
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A cubic graph G is uniquely edge-3-colorable if G has precisely one 1-factorization. S. Fiorini and R. J. Wilson [Proc. 5th Br. Comb. Conf., Aberdeen 1975, 193-202 (1976; Zbl 0345.05112)] conjectured that every uniquely edge-3-colorable planar graph must have a triangle. In this paper we show that a minimal counterexample to this conjecture is cycl...
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Let Ωn be the set of all n × n doubly stochastic matrices, let Jn be the n × n matrix all of whose entries are 1/n and let σ k (A) denote the sum of the permanent of all k × k submatrices of A. It has been conjectured that if A ε Ω n and A ≠ JJ then gA,k (θ) − σ k ((1 θ)Jn 1 θA) is strictly increasing on [0,1] for k = 2,3,…,n. We show that if A = A...
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Let (n, τ) be the set of all matrices of 0′s and 1′s of order n with exactly τ 0′s. We obtain an upper bound for the permanent of a matrix in (n, τ). For 0⩽τ⩽2n and for n2 − 2n⩽τ⩽n2 − n we determine all matrices in (n, τ) with maximum permanent.
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We define the Laplacian ratio of a tree π(T), to be the permanent of the Laplacian matrix of T divided by the product of the degrees of the vertices. Best possible lower and upper bounds are obtained for π(T) in terms of the size of the largest matching in T.
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Let G be a graph whose vertices have positive degrees d1, …, dn, respectively, and let L(G) be the Laplacian matrix of G. When G is a tree or a bipartite graph we obtain bounds for the permanent of L(G) both in terms of n only and in terms of d1, …, dn. Improved bounds are obtained in terms of the diameter of T and the size of a matching in T.
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Abstract Each vertex in a simple graph is in one of two states: \on" or \ofi". The set of all on vertices is called a conflguration. In the æ-game, \pushing" a vertex v changes the state of all vertices in the open neighborhood of v, while in the æ,-game. Keywords and Phrases: automaton, lamp lighting problem, line graph,
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A circuit is a connected 2-regular graph. A cycle is a graph such that the degree of each vertex is even. A graph G is Hamiltonian if it has a spanning circuit, and Hamiltonian-connected if for every pair of distinct vertices u, v ∈ V (G), G has a spanning (u, v)-path. A graph G is s-Hamiltonian if for any S ⊆ V (G) of order at most s, G − S has a...
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Typescript. Thesis (Ph. D.)--University of Wisconsin--Madison, 1983. Vita. Includes bibliographical references (leaves 149-151).

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