Dwight Duffus

Learn More
This paper concerns an intriguing conjecture involving vertex colorings of products of graphs. First we specify the product. Given (irreflexive, symmetric) graphs G and H, the product G x H has vertex set V(G) X V ( H ) and edges all pairs {(g, h), (g‘, h’)} such that gg‘ and hh’ are edges of G and H, respectively. We avoid modifying the label “product,” in(More)
The following equivalent results in the Boolean lattice 2 are proven. (a) Every fibre of 2 contains a maximal chain. (b) Every cutset of 2 contains a maximal antichain. (c) Every red-blue colouring of the vertices of 2 produces either a red maximal chain or a blue maximal antichain. (d) Given any n antichains in 2 there is a disjoint maximal antichain.(More)
We discuss Hedetniemi's conjecture in the context of categories of relational structures under homomorphisms. In this language Hedetniemi's conjecture says that if there are no homomorphisms from the graphs G and H to the complete graph on n vertices then there is no homomorphism from G x H to the complete graph. If an object in some category has just this(More)
Let (G, <) be a finite graph G with a linearly ordered vertex set V. We consider the decision problem (G, <)ORD to have as an instance an (unordered) graph r and as a question whether there exists a linear order < on V(T) and an order preserving graph isomorphism of (G, <) onto an induced subgraph of r. Several familiar classes of graph are characterized as(More)
A Jbre F of a partially ordered set P is a subset which intersects each nontrivial maximal antichain of P. Let I be the smallest constant such that each finite partially ordered set P contains a fibre of size at most I ‘1 PI. We show that 1, < 3 by finding a good 3-coloring of the nontrivial antichains of P. Some decision problems involving iibres are also(More)
Let L be a finite distributive lattice, and let J(L) denote the set of all join-irreducible elements of L. Set j (L) = IJ(Z)l. For each a C J(L), let u(a) denote the number of elements in the prime filter {x C L: x >~a}. Our main theorem is Theorem 1. For any finite distributive lattice L, 4 "(a) ~>j(L)41q ,'2. aEJ(L) The base 4 here can most likely be(More)
Denote by G and D, respectively, the the homomorphism poset of the finite undirected and directed graphs, respectively. A maximal antichain A in a poset P splits if A has a partition (B, C) such that for each p ∈ P either b ≤P p for some b ∈ B or p ≤p c for some c ∈ C. We construct both splitting and non-splitting infinite maximal antichains in G and in D.(More)