Haidong Wu

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 deletions of largest bonds in graphs

A well-known conjecture of Scott Smith is that any two distinct longest cycles of a k-connected graph must meet in at least k vertices when k>=2. We provide a dual version of this conjecture for two distinct largest bonds in a graph. This dual conjecture is established for k=<6.

Clonal sets in GF(q)-representable matroids

Whittle [12] conjectured that if MM is a 33-connected quaternary matroid with a clonal pair {e,f}{e,f}, then M∖e,fM∖e,f and M/e,fM/e,f are both binary. In this paper we show that for q∈{4,5,7,8,9}q∈{4,5,7,8,9} if MM is a 3-connected GF(q)GF(q)-representable matroid with a clonal set XX of size q−2q−2, then M∖XM∖X and M/XM/X are binary.

Bounding the coefficients of the characteristic polynomials of simple binary matroids

Abstract We give an upper bound and a class of lower bounds on the coefficients of the characteristic polynomial of a simple binary matroid. This generalizes the corresponding bounds for graphic matroids of Li and Tian (1978) [3] , as well as a matroid lower bound of Bjorner (1980) [1] for simple binary matroids. As the flow polynomial of a graph G is the characteristic polynomial of the dual matroid M ∗ ( G ) , the bound applies to flow polynomials.

On the circuit-spectrum of binary matroids

Murty, in 1971, characterized the connected binary matroids with all circuits having the same size. We characterize the connected binary matroids with circuits of two different sizes, where the largest size is odd. As a consequence of this result we obtain both Murtys result and other results on binary matroids with circuits of only two sizes. We also show that it will be difficult to complete the general case of this problem.

Clonal sets in GF(q)GF(q)-representable matroids

Whittle [12] conjectured that if MM is a 33-connected quaternary matroid with a clonal pair {e,f}{e,f}, then M∖e,fM∖e,f and M/e,fM/e,f are both binary. In this paper we show that for q∈{4,5,7,8,9}q∈{4,5,7,8,9} if MM is a 3-connected GF(q)GF(q)-representable matroid with a clonal set XX of size q−2q−2, then M∖XM∖X and M/XM/X are binary.

Clonal sets in G F ( q ) -representable matroids

Whittle [12] conjectured that if MM is a 33-connected quaternary matroid with a clonal pair {e,f}{e,f}, then M∖e,fM∖e,f and M/e,fM/e,f are both binary. In this paper we show that for q∈{4,5,7,8,9}q∈{4,5,7,8,9} if MM is a 3-connected GF(q)GF(q)-representable matroid with a clonal set XX of size q−2q−2, then M∖XM∖X and M/XM/X are binary.

On Tutte polynomial uniqueness of twisted wheels

A graph G is called T-unique if any other graph having the same Tutte polynomial as G is isomorphic to G. Recently, there has been much interest in determining T-unique graphs and matroids. For example, de Mier and Noy [A. de Mier, M. Noy, On graphs determined by their Tutte polynomials, Graphs Combin. 20 (2004) 105-119; A. de Mier, M. Noy, Tutte uniqueness of line graphs, Discrete Math. 301 (2005) 57-65] showed that wheels, ladders, Mobius ladders, square of cycles, hypercubes, and certain class of line graphs are all T-unique. In this paper, we prove that the twisted wheels are also T-unique.

An Extremal Problem on Contractible Edges in 3-Connected Graphs

An edge e in a 3-connected graph G is contractible if the contraction G/e is still 3-connected. The existence of contractible edges is a very useful induction tool. Let G be a simple 3-connected graph with at least five vertices. Wu [7] proved that G has at most vertices that are not incident to contractible edges. In this paper, we characterize all simple 3-connected graphs with exactly vertices that are not incident to contractible edges. We show that all such graphs can be constructed from either a single vertex or a 3-edge-connected graph (multiple edges are allowed, but loops are not allowed) by a simple graph operation.

Distribution of contractible elements in 2-connected matroids

An element e in a 2-connected matroid M is contractible if its contraction M/e is 2-connected. The existence of contractible elements provides a very useful induction tool. In this paper, we study the distribution of contractible elements in simple 2-connected matroids.