Talmage James Reid

Ramsey Numbers for Matroids

Letkandlbe positive integers. The Ramsey numberr(k,l)is the least positive integerrsuch that every graphGwithrvertices contains eitherkmutually adjacent vertices orlmutually non-adjacent vertices. A matroid version of the Ramsey number is defined in this paper. Results which are strikingly similar to the classical graph theorems are obtained. For example, the upper bound of (k+l-2k-1)forr(k,l)of Erdos and Szekeres has an analogue in the matroid Ramsey numbers considered here.

Some Small Circuit-Cocircuit Ramsey Numbers for Matroids

Ramsey numbers for matroids, which mimic properties of Ramsey numbers for graphs, have been denned as follows. Let k and l be positive integers. Then n ( k, l ) is the least positive integer n such that every connected matroid with n elements contains either a circuit with at least k elements or a cocircuit with at least l elements. We determine the largest known value of these numbers in the sense of maximizing both k and l . We also find extremal matroids with small circuits and cocircuits. Results on matroid connectivity, geometry, and extremal matroid theory are used here.

The Matroid Ramsey Number n (6,6)

A tight upper bound on the number of elements in a connected matroid with fixed rank and largest cocircuit size is given. This upper bound is used to show that a connected matroid with at least thirteen elements contains either a circuit or a cocircuit with at least six elements. In the language of matroid Ramsey numbers, n(6, 6) = 13: this is the largest currently known matroid Ramsey number.

Degree sequence and independence in K (4)-free graphs

Abstract We investigate whether K r -free graphs with few repetitions in the degree sequence may have independence number o( n ). We settle the cases r = 3 and r ⩾ 5, and give partial results for the very interesting case r = 4.

Triangles in 3-connected matroids

A collection F of 3-connected matroids is triangle-rounded if, whenever M is a 3-connected matroid having a minor in F, and T is a 3-element circuit of M, then M has a minor which uses T and is isomorphic to a member of F. An efficient theorem for testing a collection of matroids for this property is presented. This test is used to obtain several results including the following extension of a result of Asano, Nishizeki, and Seymour. Let T be a 3-element circuit of a 3-connected binary nonregular matroid M with at least eight elements. Then M has a minor using T that is isomorphic to S8 or the generalized parallel connection across T of F7 and M(K4).

Obstructions to a binary matroid being graphic

Bixby and Cunningham showed that a 3-connected binary matroid M is graphic if and only if every element belongs to at most two non-separating cocircuits. Likewise, Lemos showed that such a matroid M is graphic if and only if it has exactly r(M)+1 non-separating cocircuits. Hence the presence in M of either an element in at least three non-separating cocircuits, or of at least r(M)+2 non-separating cocircuits, implies that M is non-graphic. We provide lower bounds on the size of the set of such elements, and on the number of non-separating cocircuits, in such non-graphic binary matroids. A computationally efficient method for finding such lower bounds for specific minor-closed classes of matroids is given. Applications of this method and other results on sets of obstructions to a binary matroid being graphic are given.

On Minimally 3-Connected Binary Matroids

We generalize a minimal 3-connectivity result of Halin from graphs to binary matroids. As applications of this theorem to minimally 3-connected matroids, we obtain new results and short inductive proofs of results of Oxley and Wu. We also give new short inductive proofs of results of Dirac and Halin on minimally k-connected graphs for k ∈ {2,3}.

Sizes of graphs with induced subgraphs of large maximum degree

Abstract Graphs with n + k vertices in which every set of n + j vertices induce a subgraph of maximum degree at least n are considered. For j = 1 and for k fairly small compared to n , we determine the minimum number of edges in such graphs.

Note: A note on clone sets in representable matroids

We investigate the size of clone sets in representable matroids.

On elements in small cocircuits in minimally k -connected graphs and matroids

We give a lower bound on the number of edges meeting some vertex of degree k in terms of the total number of edges in a minimally k-connected graph. This lower bound is tight if k is two or three. The extremal graphs in the case that k=2 are characterized. We also give a lower bound on the number of elements meeting some 2-element cocircuit in terms of the total number of elements in a minimally 2-connected matroid. This lower bound is tight and the extremal matroids are characterized.