Robert E. Woodrow

On monochromatic paths in edge-coloured digraphs☆

Abstract Let G be a directed graph whose edges are coloured with two colours. Call a set S of vertices of Gindependent if no two vertices of S are connected by a monochromatic directed path. We prove that if G contains no monochromatic infinite outward path, then there is an independent set S of vertices of G such that, for every vertex x not in S, there is a monochromatic directed path from x to a vertex of S. In the event that G is infinite, the proof uses Zorns lemma. The last part of the paper is concerned with the case when G is a tournament.

Cycles and rays

Linkability in Countable-Like Webs.- Decomposition into Cycles I: Hamilton Decompositions.- An Order- and Graph- Theoretical Characterisation of Weakly Compact Cardinals.- Small Cycle Double Covers of Graphs.- ?-Transformations, Local Complementations and Switching.- Two Extremal Problems in Infinite Ordered Sets and Graphs: Infinite Versions of Menger and Gallai-Milgram Theorems for Ordered Sets and Graphs.- Chvatal-Erd?s Theorem for Digraphs.- Long Cycles and the Codiameter of a Graph II.- Compatible Euler Tours in Eulerian Digraphs.- A.J.W. Hilton, C.A. Rodger, Edge-Colouring Graphs and Embedding Partial Triple Systems of Even Index.- On the Rank of Fixed Point Sets of Automorphisms of Free Groups.- On Transition Polynomials of 4-Regular Graphs.- On Infinite n-Connected Graphs.- Ordered Graphs Without Infinite Paths.- Ends of Infinite Graphs, Potential Theory and Electrical Networks.- Topological Aspects of Infinite Graphs.- Dendroids, End-Separators, and Almost Circuit-Connected Trees.- Partition Theorems for Graphs Respecting the Chromatic Number.- Vertex-Transitive Graphs That Are Not Cayley Graphs.

Two-colouring all two-element maximal antichains

Abstract A fibre in a partially ordered set P is a subset of P meeting every maximal antichain of P . We give an example of a finite poset P with no one-element maximal antichain and containing no fibre of size at most | P }2, thus answering a question of Aigner-Andreae and disproving a conjecture of Lonc-Rival. We also prove Theorem 1. The elements of an arbitrary partially ordered set can be coloured with two colours such that every two-element maximal antichain receives both colours .

Lexicographic matchings cannot form Hamiltonian cycles

For any positive integer k let B(k) denote the bipartite graph of k- and k+1-element subsets of a 2k+1-element set with adjacency given by containment. It has been conjectured that for all k, B(k) is Hamiltonian. Any Hamiltonian cycle would be the union of two (perfect) matchings. Here it is shown that for all k>1 no Hamiltonian cycle in B(k) is the union of two lexicographic matchings.

Absorbing sets in arc-coloured tournaments

Let T be a tournament whose arcs are coloured with k colours. Call a subset X of the vertices of T absorbing if from each vertex of T not in X there is a monochromatic directed path to some vertex in X. We consider the question of the minimum size of absorbing sets, extending known results and using new approaches. The greater part of the paper deals with finite tournaments, the last section treats infinite ones. In each case questions are suggested, both old and new.

On cop-win graphs☆

Abstract Following a question of Anstee and Farber we investigate the possibility that all bridged graphs are cop-win. We show that infinite chordal graphs, even of diameter two, need not be cop-win and point to some interesting questions, some of which we answer.

Finite cutsets and finite antichains

An ordered set (P,≤) has the m cutset property if for each x there is a set Fx with cardinality less than m, such that each element of Fx is incomparable to x and {x} ∪ Fx meets every maximal chain of (P,≤). Let n be least, such that each element x of any P having the m cutset property belongs to some maximal antichain of cardinality less than n. We specify n for m < w. Indeed, n-1=m= width P for m=1,2,n=5 if m=3 and n⩾ℵ1 if m ≥4. With the added hypothesis that every bounded chain has a supremum and infimum in P, it is shown that for 4⩽m⩽ℵ0, n=ℵ0. That is, if each element x has a finite cutset Fx, each element belongs to a finite maximal antichain.

Enumeration of order preserving maps

Three results are obtained concerning the number of order preserving maps of an n-element partially ordered set to itself. We show that any such ordered set has at least 22n/3 order preserving maps (and 22 in the case of length one). Precise asymptotic estimates for the numbers of self-maps of crowns and fences are also obtained. In addition, lower bounds for many other infinite families are found and several precise problems are formulated.

There are four countable ultrahomogeneous graphs without triangles

Let G be a countably infinite ultrahomogeneous undirected graph in which the complete graph on three vertices K3 cannot be embedded. Then G is isomorphic to one of the following four graphs: 1. (i) the countable graph on ω with no edges; 2. (ii) the graph 〈ω, V〉 with V = {(2n, 2n + 1): n ϵ w} U{(2n + 1, 2n): nϵ w} 3. (iii) the graph 〈ω to, W〉 where W {(i, j) : i + j is odd}; or 4. (iv) the graph G3, is a graph universal for the class of countably infinite graphs omitting K3.

Theories with a Finite Number of Countable Models

We give two examples. T 0 has nine countable models and a nonprincipal 1-type which contains infinitely many 2-types. T 1 , has four models and an inessential extension T 2 having infinitely many models.