James G. Oxley

On weakly symmetric graphs of order twice a prime

Abstract A graph is weakly symmetric if its automorphism group is both vertex-transitive and edge-transitive. In 1971, Chao characterized all weakly symmetric graphs of prime order and showed that such graphs are also transitive on directed edges. In this paper we determine all weakly symmetric graphs of order twice a prime and show that these graphs too are directed-edge transitive.

Typical subgraphs of 3- and 4-connected graphs

Abstract We prove that, for every positive integer k , there is an integer N such that every 3-connected graph with at least N vertices has a minor isomorphic to the k -spoke wheel or K 3, k ; and that every internally 4-connected graph with at least N vertices has a minor isomorphic to the 2 k -spoke double wheel, the k -rung circular ladder, the k -rung Mobius ladder, or K 4, k . We also prove an analogous result for infinite graphs.

On Inequivalent Representations of Matroids over Finite Fields

Kahn conjectured in 1988 that, for each prime powerq, there is an integern(q) such that no 3-connectedGF(q)-representable matroid has more thann(q) inequivalentGF(q)-representations. At the time, this conjecture was known to be true forq=2 andq=3, and Kahn had just proved it forq=4. In this paper, we prove the conjecture forq=5, showing that 6 is a sharp value forn(5). Moreover, we also show that the conjecture is false for all larger values ofq.

On the Structure of 3-connected Matroids and Graphs

An element e of a 3 -connected matroid M is essential if neither the deletionM\e nor the contraction M/e is 3 -connected. Tutte?s Wheels and Whirls Theorem proves that the only 3 -connected matroids in which every element is essential are the wheels and whirls. In this paper, we consider those 3 -connected matroids that have some non-essential elements, showing that every such matroid M must have at least two such elements. We prove that the essential elements of M can be partitioned into classes where two elements are in the same class if M has a fan, a maximal partial wheel, containing both. We also prove that if an essential element e of M is in more than one fan, then that fan has three or five elements; in the latter case, e is in exactly three fans. Moreover, we show that if M has a fan with 2 k or 2 k+ 1 elements for some k? 2, then M can be obtained by sticking together a (k+ 1)-spoked wheel and a certain 3 -connected minor of M. The results proved here will be used elsewhere to completely determine all 3 -connected matroids with exactly two non-essential elements.

The structure of the 3-separations of 3-connected matroids

Abstract Tutte defined a k -separation of a matroid M to be a partition ( A , B ) of the ground set of M such that | A |,| B |⩾ k and r ( A )+ r ( B )− r ( M ) k . If, for all m n , the matroid M has no m -separations, then M is n -connected. Earlier, Whitney showed that ( A , B ) is a 1-separation of M if and only if A is a union of 2-connected components of M . When M is 2-connected, Cunningham and Edmonds gave a tree decomposition of M that displays all of its 2-separations. When M is 3-connected, this paper describes a tree decomposition of M that displays, up to a certain natural equivalence, all non-trivial 3-separations of M .

The binary matroids with no 4-wheel minor

On caracterise les matroides binaires sans mineurs isomorphes a une roue de rang 4. On resout pour cette classe de matrices le probleme critique et on etend les resultats de Kung-Walton-Welsh pour les classes associees de matroides binaires

On nonbinary 3-connected matroids

It is well known that a matroid is binary if and only if it has no minor isomorphic to U2,4, the 4-point line. Extending this result, Bixby proved that every element in a nonbinary connected matroid is in a U2,4minor. The result was further extended by Seymour who showed that every pair of elements in a nonbinary 3-connected matroid is in a U2,4-minor. This paper extends Seymours theorem by proving that if {x, y, z} is contained in a nonbinary 3-connected matroid M, then either M has a U2,4-minor using {x, y, z}, or M has a minor isomorphic to the rank-3 whirl that uses {x, y, z} as its rim or its spokes.

Tutte polynomials computable in polynomial time

Abstract We show that for any accessible class of matroids of bounded width, the Tutte polynomial is computable in polynomial time.

Unavoidable Minors of Large 3-Connected Matroids

This paper proves that, for every integernexceeding two, there is a numberN(n) such that every 3-connected matroid with at leastN(n) elements has a minor that is isomorphic to one of the following matroids: an (n+2)-point line or its dual, the cycle or cocycle matroid ofK3,n, the cycle matroid of a wheel withnspokes, a whirl of rankn, or ann-spike. A matroid is of the last type if it has ranknand consists ofnthree-point lines through a common point such that, for allkin {1,2,?,n?1}, the union of every set ofkof these lines has rankk+1.

Unavoidable Minors of Large 3-Connected Binary Matroids

We show that, for every integerngreater than two, there is a numberNsuch that every 3-connected binary matroid with at leastNelements has a minor that is isomorphic to the cycle matroid ofK3,n, its dual, the cycle matroid of the wheel withnspokes, or the vector matroid of the binary matrix (In|Jn?In), whereJnis then×nmatrix of all ones.