John L. Goldwasser

Fibonacci Polynomials and Parity Domination in Grid Graphs

Abstract. 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 reflexive). 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 in grid graphs and properties of a corresponding recurrence.

Characterizing switch-setting problems ∗

Algebraic conditions and algorithmic procedures are given to determine whether an m × 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 finite field representation of the problem, and an analysis of Fibonacci polynomials, conditions on m and n are given which characterize when the m × n problem can be solved.

Extending the disjoint-representatives theorems of Hall, Halmos, and Vaughan to list-multicolorings of graphs

A multigraph M with maximum degree Δ(M) is called critical, if the chromatic index χ2(M) > Δ(M) and χ2(M - e) = χ2(M) - 1 for each edge e of M. The weak critical graph conjecture [1, 7] claims that there exists a constant c > 0 such that every critical multigraph M with at most c · Δ(M) vertices has odd order. We disprove this conjecture by constructing critical multigraphs of order 20 with maximum degree k for all k e 5.

Maximum permanents of matrices of zeros and ones

Abstract Let U (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 U (n, τ). For 0⩽τ⩽2n and for n2 − 2n⩽τ⩽n2 − n we determine all matrices in U (n, τ) with maximum permanent.

The density of ones in Pascal's rhombus

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. c 1999 Elsevier Science B.V. All rights reserved

Odd and residue domination numbers of a graph

Let G = (V, E) be a simple, undirected graph. A set of vertices D 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 set is called the odd domination number of G, denoted by γ1(G). In this paper, several algorithmic and structural results are presented on this parameter for grids, complements of powers of cycles, and other graph classes as well as for more general forms of “residue” domination.

Monotonicity of permanents of direct sums of doubly stochastic matrices

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 e 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 1 ⊕ ⊕At (t ≥ 2) is an n × n matrix where Ai for i = 1,2, …,t, and if for each i gAi,ki (θ) is non-decreasing on [0.1] for kt = 2,3,…,ni , then gA,k (θ) is strictly increasing on [0,1] for k = 2,3,…,n.

Permanent of the Laplacian matrix of trees with a given matching

Abstract 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 .

Maximum Orbit Weights in the σ-game and Lit-only σ-game on Grids and Graphs

Abstract 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 smallest possible number of on vertices. We show that any starting configuration in a graph with no isolated vertices can, by a sequence of pushes, be reduced to at most half on, and we characterize those graphs for which you cannot do better. The proofs use techniques from coding theory. In the lit-only versions of these two games, you can only push vertices which are on. We obtain some results on the minimum number of on vertices one can obtain in grid graphs in the regular and lit-only versions of both games.

Uniquely Edge-3-Colorable Graphs and Snarks

Abstract. 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-colorable cubic graph must have the Petersen graph as a minor. Fiorini and Wilson (1976) conjectured that every uniquely edge-3-colorable planar cubic graph must have a triangle. It is proved in this paper that every counterexample to this conjecture is cyclically 5-edge-connected and that in a minimal counterexample to the conjecture, every cyclic 5-edge-cut is trivial (an edge-cut T of G is cyclic if no component of G\T is acyclic and a cyclic edge-cut T is trivial if one component of G\T is a circuit of length |T|).