William F. Klostermeyer

(2 + ε)-Coloring of planar graphs with large odd-girth

Let G be a 2-connected graph, let u and v be distinct vertices in V(G), and let X be a set of at most four vertices lying on a common (u, v)-path in G. If deg(x) ≥ d for all x e V(G) \ {u, v}, then there is a (u, v)-path P in G with X ‚ V(P) and |E(P)| ≥ d.

Homomorphisms and oriented colorings of equivalence classes of oriented graphs

We combine two topics in directed graphs which have been studied separately, vertex pushing and homomorphisms, by studying homomorphisms of equivalence classes of oriented graphs under the push operation. Some theory of these mappings is developed and the complexity of the associated decision problems is determined. These results are then related to oriented colorings. Informally, the pushable chromatic number of an oriented graph G is the minimum value of the oriented chromatic number of any digraph obtainable from G using the push operation. The pushable chromatic number is used to give tight upper and lower bounds on the oriented chromatic number. The complexity of deciding if the pushable chromatic number of a given oriented graph is at most a fixed positive integer k is determined. It is proved that the pushable chromatic number of a partial 2-tree is at most four. Finally, the complexity of deciding if the oriented chromatic number of a given oriented graph is at most a fixed positive integer k is determined.

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.

Secure domination and secure total domination in graphs

A secure (total) dominating set of a graph G = (V; E) is a (total) dominating set X V with the property that for each u 2 V X, there exists x 2 X adjacent to u such that (X fxg) [ fug is (total) dominating. The smallest cardinality of a secure (total) dominating set is the secure (total) domination number s(G) ( st(G)). We characterize graphs with equal total and secure total domination numbers. We show that if G has minimum degree at least two, then st(G) s(G). We also show that st(G) is at most twice the clique covering number of G, and less than three times the independence number. With the exception of the independence number bound, these bounds are sharp.

Graphs with equal eternal vertex cover and eternal domination numbers

Mobile guards on the vertices of a graph are used to defend it against an infinite sequence of attacks on either its vertices or its edges. If attacks occur at vertices, this is known at the eternal domination problem. If attacks occur at edges, this is known as the eternal vertex cover problem. We focus on the model in which all guards can move to neighboring vertices in response to an attack. Motivated by the question of which graphs have equal eternal vertex cover and eternal domination numbers, a number of results are presented; one of the main results of the paper is that the eternal vertex cover number is greater than the eternal domination number (in the all-guards move model) in all graphs of minimum degree at least two.

Analogues of cliques for oriented coloring

We examine subgraphs of oriented graphs in the context of oriented coloring that are analogous to cliques in traditional vertex coloring. Bounds on the sizes of these subgraphs are given for planar, outerplanar, and series-parallel graphs. In particular, the main result of the paper is that a planar graph cannot contain an induced subgraph D with more than 36 vertices such that each pair of vertices in D are joined by a directed path of length at most two.

n -Tuple Coloring of Planar Graphs with Large Odd Girth

Abstract. The main result of the papzer is that any planar graph with odd girth at least 10k−7 has a homomorphism to the Kneser graph Gk2k+1, i.e. each vertex can be colored with k colors from the set {1,2,…,2k+1} so that adjacent vertices have no colors in common. Thus, for example, if the odd girth of a planar graph is at least 13, then the graph has a homomorphism to G25, also known as the Petersen graph. Other similar results for planar graphs are also obtained with better bounds and additional restrictions.

Hamiltonicity and reversing arcs in digraphs

We give a simple proof that the obvious necessary conditions for a graph to contain the kth power of a Hamiltonian path are sufficient for the class of interval graphs. The proof is based on showing that a greedy algorithm tests for the existence of Hamiltonian path powers in interval graphs. We will also discuss covers by powers of paths and analogues of the Hamiltonian completion number.

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