Richard P. Anstee

Characterizations of totally balanced matrices

Abstract A (0,1)-matrix is totally balanced if it does not contain as a submatrix the incidence matrix of any cycle of length at least 3. Several alternative characterizations of these matrices are presented. These characterizations follow from properties of strongly chordal graphs, studied by Farber, and maximal totally balanced matrices, studied by Anstee. Using these characterizations, efficient recognition algorithms for totally balanced matrices are presented. In addition, a new completion algorithm for building a maximal totally balanced matrix from an arbitrary totally balanced matrix follows from these results.

A polynomial algorithm for b-matchings: an alternative approach

Abstract A b-matching of a given graph G is an assignment of integer weights to the edges of G so that the sum of the weights on the edges incident with a vertex v is at most bv (b denotes the vectors of bvs). When bv = 1 for all vertices v in G, then b-matchings are the usual matchings. The b-matching problem asks for a b-matching of maximum cost where the edges of G have been assigned costs and the cost of a b-matching is the sum of the weights times the costs. We do not assume G to be bipartite. We present a polynomial algorithm for the b-matching problem differing from the algorithm of Cunningham and Marsh (1979). In fact, the new algorithm is strongly polynomial using the new strongly polynomial minimum cost flow algorithm of Tardos (we actually use the algorithm of Orlin (1984)) to obtain a half-integral optimal solution. An integral solution (close to being optimal) is obtained from it which is used as input to Pulleyblanks (1973) b-matching algorithm.

An algorithmic proof of Tutte's f-factor theorem

Abstract Let G be a multigraph on n vertices, possibly with loops. An f-factor is a subgraph of G with degree fi at the ith vertex for i = 1, 2,…, n. Tuttes f-factor theorem is proved by providing an algorithm that either finds an f-factor or shows that it does not exist and does this in O(n3) operations. Note that the complexity bound is independent of the number of edges of G and the degrees fi. The algorithm is easily altered to handle the problem of looking for a symmetric integral matrix with given row and column sums by assigning loops degree one. A (g,f)-factor is a subgraph of G with degree di at the ith vertex, where gi ⩽ di ⩽ fi, for i = 1,2,…, n. Lovaszs (g,f)-factor theorem is proved by providing an O(n3) algorithm to either find a (g,f)-factor or show one does not exist.

Knot polynomials and generalized mutation

Abstract The motivation for this work was to construct a nontrivial knot with trivial Jones polynomial. Although that open problem has not yielded, the methods are useful for other problems in the theory of knot polynomials. The subject of the present paper is a generalization of Conways mutation of knots and links. Instead of flipping a 2-strand tangle, one flips a many-string tangle to produce a generalized mutant. In the presence of rotational symmetry in that tangle, the result is called a “rotant”. We show that if a rotant is sufficiently simple, then its Jones polynomial agrees with that of the original link. As an application, this provides a method of generating many examples of links with the same Jones polynomial, but different Alexander polynomials. Various other knot polynomials, as well as signature, are also invariant under such moves, if one imposes more stringent conditions upon the symmetries. Applications are also given to polynomials of satellites and symmetric knots.

The network flows approach for matrices with given row and column sums

We study the set of integral matrices with given row and column sums where each position has a bound on the size of the entry in it. Such matrices correspond to maximum integral flows in certain networks. The well known existence theorem follows from the max flow-min cut theorem. A general network flow result, specialized to our setting, yields a useful interchange theorem which has a number of corollaries. Prompted by another network flow result, new results are obtained for invariant positions i.e. entries which are the same regardless of which matrix is chosen. This leads to a classification of invariant edges in graphs of a given degree sequence, important in the study of split graphs and threshold graphs.

Planes of order 10 do not have a collineation of order 5

The existence of a projective plane of order 10 remains in doubt. If one does exist it may have only the identity collineation. D. R. Hughes [I, 2] showed that for a plane of order n where y1 = 2 (mod 4) and IZ > 2 the collineation group is of odd order. He also showed that for a plane of order IO the only primes dividing the order of the collineation group could be 3, 5, or 11, that for order 3 there would be 3 or 9 fixed points (also lines) and for 5 exactly one fixed point and one fixed line. Whitesides [9] has eliminated the possibility of a collineation of order Il. She has also [lo] eliminated orders 9,25, and 15, so that the only remaining possible orders are 1, 3, or 5. The main result of this paper is to eliminate the order 5.

Properties of (0, 1)-matrices with no triangles ☆

Abstract We study (0, 1)-matrices which contain no triangles (submatrices of order 3 with row and column sums 2) previously studied by Ryser. Let the row intersection of row i and row j of some matrix, when regarded as a vector, have a 1 in a given column if both row i and row j do and a zero otherwise. For matrices with no triangles, columns sums ⩾2, we find that the number of linearly independent row intersections is equal to the number of distinct columns. We then study the extremal (0, 1)-matrices with no triangles, column sums ⩾2, distinct columns, i.e., those of size mx(m2). The number of columns of column sum l is m − l + 1 and they form a (l − 1)-tree. The ((m2)) columns have a unique SDR of pairs of rows with 1s. Also, these matrices have a fascinating inductive buildup. We finish with an algorithm for constructing these matrices.

Simplified existence theorems for (g,f)-factors

Abstract A ( g , f )-factor of a graph G is a subgraph of G whose valencies are bounded between g and f . Let G =( V , E ) and let g =( g v : v ∈ V ), f =( f v : v ∈ V ) be vectors of integers with 0≤ g v ≤ f v ≤deg G ( v ). Then a ( g , f )-factor is a subgraph H of G so that g v ≤deg H ( V )≤ f v . The conditions forthe existence of a ( g , f )- factor due to Lovasz involve considering all disjoint pairs of subsets ofvertices S , T in contrast to Tuttes conditions for a perfect matching which considers a single subset S of vertices. We seek conditions for the existence of ( g , f )-factors of this latter type and extend results of Heinrich, Hell, Liu and Kirkpatrick, and Hell and Kirkpatrick and others. The technique is to consider fractional ( g , f )-factors (whose existence conditions are of the desired single-subset type) and use alternating walks to seek a ( g , f )-factor starting from the fractional factor. Certain conditions on G , g , f restrict the form of a barrier to ( g , f )-factor.

Small Forbidden Configurations

In the present paper we continue the work begun by Sauer, Perles, Shelah and Anstee on forbidden configurations of 0–1 matrices. We give asymptotically exact bounds for all possible 2 × l forbidden submatrices and almost all 3 × l ones. These bounds are improvements of the general bounds, or else new constructions show that the general bound is best possible. It is interesting to note that up to the present state of our knowledge every forbidden configuration results in polynomial asymptotic.

Small Forbidden Configurations IV: The 3 Rowed Case

The present paper continues the work begun by Anstee, Griggs and Sali on small forbidden configurations. We define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. Let F be a k×l (0,1)-matrix (the forbidden configuration). Small refers to the size of k and in this paper k = 3. Assume A is an m×n simple matrix which has no submatrix which is a row and column permutation of F. We define forb(m,F) as the best possible upper bound on n, for such a matrix A, which depends on m and F. We complete the classification for all 3-rowed (0,1)-matrices of forb (m,F) as either Θ(m), Θ(m2) or Θ(m3) (with constants depending on F).