Showing papers in "The Journal of Combinatorics in 1987"

The combinatorics of q -Hermite polynomials and the Askey-Wilson integral

Abstract: The q-Hermite polynomials are defined as a q-analogue of the matching polynomial of a complete graph. This allows a combinatorial evaluation of the integral used to prove the orthogonality of Askey and Wilson's 4φ3 polynomials. A special case of this result gives the linearization formula for q-Hermite polynomials. The moments and associated continued fraction are explicitly given. Another set of polynomials, closely related to the q-Hermite, is defined. These polynomials have a combinatorial interpretation in terms of finite vector spaces which give another proof of the linearization formula and the q-analogue of Mehler's formula.

Generalized quadrangles and flocks of cones

Abstract: A flock of the quadratic cone K of PG(3, q) is a partition of K but its vertex into disjoint conics. If the planes of the q conics of such a flock all contain a common line, then the flock is called linear. With any flock there corresponds a translation plane which is Desarguesian iff the flock is linear. W. M. Kantor showed that with a set of q upper triangular 2 × 2-matrices over GF(q) of a certain type, there corresponds a generalized quadrangle of order (q2, q). We prove that with such a set of q matrices there corresponds a flock of the quadratic cone of PG(3, q), and conversely that with each flock of the quadratic cone there corresponds such a set of matrices. Using this relationship, new flocks, new generalized quadrangles, and probably new translation planes are obtained.

Isomorphism problem for relational structures with a cyclic automorphism

Abstract: The problem we discuss here is the conjecture of Adam [1] concerning the isomorphism problem of cyclic (or, in other words, circulant) graphs and its generalization by Babai [4]. We attempt to give a fairly complete survey of results related to Adam's conjecture. We prove that the isomorphism classes of all n -element relational structures (not only graphs) with cyclic automorphism have the anticipated simple form if and only if ( n , ϕ( n )) = 1 or n = 4, where ϕ( n ) is Euler's ϕ function. The same result holds if only quaternary relational structures are considered.

A new property of critical imperfect graphs and some consequences

Abstract: We prove a new property of critical imperfect graphs. As a consequence, we define a new class of perfect graphs. This class contains perfectly orderable graphs and graphs in which that every odd cycle has two chords.

Partitions with prescribed hook differences

Abstract: We investigate partition identities related to off-diagonal hook differences. Our results generalize previous extensions of the Rogers—Ramanujan identities. The identity of the related polynomials with constructs in statistical mechanics is discussed.

Factoring a graph in polynomial time

Abstract: We present a polynomial-time algorithm for deciding whether a given connected graph is a non-trivial Cartesian product. The method entails first representing the graph as an isometric subgraph of a Cartesian product of graphs, then finding a suitable partition of the factors.

Saturated configurations of points in projective Galois spaces

Abstract: In this paper we examine sets K of k points in a projective Galois space PG(r, q), of any dimension r, satisfying the following property: the union of all ϱ-subspaces, 0 ⩽ ϱ ⩽ r, of PG(r, q) generated by (ϱ + 1) independent points in K coincides with the whole space. Moreover, estimates for the smallest possible value of such a k are given.

On 2-transitive graphs of girth 5

Abstract: Graphs with girth 5 and automorphism group acting transitively on the set of paths of length 2 are investigated.

On a pursuit game of Cayley diagraphs

Abstract: We consider a pursuit game on Abelian Cayley digaphs. We obtain a bound for the number of pursuers necessary to catch one evader. We prove in the Abelian undirected case that ⌈ 3 k 4 Ȱ pursuers can catch the evader, where k is the degree.

A characterization of the association schemes of bilinear forms

Abstract: If n ⩾ 2d ⩾ 6 and q ⩾ 4, the distance-regular graphs Hq (n, d) (with d × n matrices over GF(q) as vertex set, and two vertices A, B being adjacent if and only if the rank of A − B is 1) are characterized by their parameters together with the property that the number of edges of the induced subgraph on the common neighborhood of vertices x and y depends only on the distance between them. As a corollary, we prove that Hq (n, d) has no antipodal covering if n ⩾ 2d ⩾ 6 and q ⩾ 4.

Arcs and blocking sets II

Abstract: In a finite projective plane 1t a blocking set is a set S of points such that each line contains at least one point in S and at least one point not in S. The main results in this note are Theorems 1.1 and 1.7 and Corollaries 1.8 and 1.9. Theorem 1.1 describes new bounds on certain kinds of reduced blocking sets in PG(2, q). Theorem 1.7 and Corollary 1.8 give new bounds on the cardinality of a reduced (and so, of an arbitrary) blocking set Sin PG(2, q). These bounds yield a significant improvement on previously known results. The proof of 1.7 uses a combinatorial argument together with special cases of some deep results in Jamison [9] and R6dei [10]. A fortuitous factorization makes the result more tractable. (The result of 1.1 is used in 1.4 to obtain bounds on the size of complete arcs in PG(2, q): in some cases these results give a slight improvement on the results in Hirschfeld [8]). Corollary 1.9 shows how the structure ofPG(2, q) is being utilized: it yields a far stronger result than a related result for general planes in Bruen and Thas [5] (the case n > 4 in Theorem 3 there). For blocking sets in arbitrary finite projective planes not much is known apart from 2.2. Here we offer a new proof based on an idea in Hill and Mason [7]. Moreover the proof can be generalized to arbitrary 2-designs as in Theorem 2.3. Fundamental to the improved bound on lSI in PG(2, q) is a result of Jamison [9] on intersection sets in the classical affine plane AG(2, q). His result is not valid for general finite affine planes: the problem of finding good bounds on the size of blocking sets in finite affine planes is open. Our result (Theorem 3.1) describes the best known such bounds. We use the fact that an affine plane (which is of course a 2-design) also has its lines arranged into parallel classes, so our result is a slight improvement on Theorem 2.3 in the case of finite affine planes.

Signed graphs represented by D

Abstract: In this paper, a characterization of the family of sigraphs represented by D n for all n is proved. The minimal forbidden sigraphs for that family are described. Using that it is shown that any minimal forbidden sigraph for the family of sigraphs with eigenvalues ≥ - 2 has at most 10 vertices. An application to quadratic forms is also given.

Root systems and the Johnson and Hamming graphs

Abstract: In [27] we show that any distance-regular graph Г containing a cycle {ν0, ν1, ν2, ν3, ν0} with δ(ν0, ν2) = δ(ν1, ν3) = 2 was finite, with diameter d, valency k and intersection numbers a1, cd satisfying d ⩽ k + c d a i + 2 with equality holding if and only if (1) c i − c i − 1 + b i − 1 − b i − a 1 − 2 = 0 , ( 1 ⩽ i ⩽ d ) In this graph we give a simplified proof of this fact, and then classify the graphs where the diameter bound is attained. Not assuming the existence of the above cycle in Γ but only that c2 > 2, we use the classification of the classical root systems to show a distance-regular graph satisfying (1) is either the Johnson graph J(d, n), a graph with the intersection numbers of the Hamming graph H(d, q), the Cocktail Party graph CP(n), ½ H(n, 2) (the halved graph of the n-cube), or one of a finite number of exceptional graphs, all with d ≤ 8, a1 ≤ 16, and k ≤ 28.

Altshuler's sphere M10-425- is not polytopal

Abstract: There is still no algorithm to decide in reasonable time whether a combinatorial sphere is polytopal or not. A short method of proof for showing that a combinatorial sphere is not polytopal is described in the case of a 3-sphere with 10 vertices (Altshuler's M 425 10 ).

Some remarks on Catalan families

Abstract: of n words of length r obtained by splitting all permutations of {O, 1, ... , rn I} into n pieces of equal length r. It suffices therefore to give a bijection between the set of all n-fold products formed from (r l)n + 1 different elements XQ, Xl' ... , x(r-l)n and the set A~. To simplify the exposition we shall write i instead of Xi and consider the usual order on the integers. Let P be any such product. It is formed using n pairs of brackets. Consider all inner brackets (i.e. those which contain no other brackets) and order them with respect to increasing largest elements. Choose the first inner bracket in this ordering and call it (r l)n + 1. Then P may be written as an (n I)-fold product of elements from a subset A £: {O, 1, ... , (r l)n + I} with (r 1)(n 1) + 1 elements. Iterating this procedure we get a uniquely determined sequence of n brackets (r l)n + 1, ... , rn. This sequence defines an element of A~.

Covering with Euclidean boxes

Abstract: Parallelopipeds of the d-dimensional Euclidean space ℝd with faces parallel to the axes are called boxes and box(p, q) denotes the intersection of all boxes containing the points p, q ∈ ℝd. Here we prove the existence of a constant c depending only on d such that any compact set V ⊂ ℝd has a subset of cardinality at most c which induces a box-cover of V, that is V ⊂ ∪p,g∈S box(p, q). That result is used to derive further covering theorems in combinatorial geometry and hypergraph theory.

Handles in graphs and sphere bundles over S 1

Abstract: We extend to (n + 1)-coloured graphs the concept of combinatorial handle, presented for n = 3, in [7], [11], [19]. Then we study the operation of cancelling of such a handle, which always reveals a connected sum decomposition of the represented manifolds.

On the product of certain permutations

Abstract: A theorem by M. Cohn and A. Lempel, relating the product of certain permutations to the rank of an associated matrix, is restated and reproved. Its relationship to old and recent results in topological graph theory is pointed out.

On the null-homotopy of bridged graphs

Abstract: A combinatorial notion of null-homotopy for graphs was introduced by Duchet, Las Vergnas, and Meyniel. Their results were of a qualitative nature for all such graphs. Here some quantitative results are established for the special class of graphs which contain no isometric cycles other than triangles. It is also shown how each cycle in such a graph may be decomposed into chordal pieces.

Perfect codes and regular partitions in graphs and groups

Abstract: We define the concept of regular partition of a graph Γ and its relationship to the automorphism group of Γ. In application, we give necessary conditions for a perfect code of Γ in the sense of Biggs to exist. We recover a theorem of C. Landauer relating perfect codes in a finite group to its characters. We establish a necessary condition on the subgroups for the existence of a perfect code. As an application, we mention a result of O. Rothaus and J. G. Thompson concerning the existence of perfect codes in permutation groups.

An axiomatic of non-Radon partitions of oriented matroids

Abstract: Let M ( E , O ) be an oriented matroid. We say that {A, E\A} is a non-Radon partition of M if O A ¯ = O E \ A ¯ is an acyclic reorientation of O . This definition generalizes the classic notion of (non)-Radon partition of a finite subset E of ℝd. We give an intrinsic characterization of the families of partitions which are the family of all non-Radon partitions of some oriented matroid.

On inversions and cycles in permutations

Abstract: There is essentially nothing known about F(x, y), with the exception of a result of Gessel [5, sect. 7] which computes the right-hand side of (1) where (1 is an involution. For reasons which we will discuss in section IV we believe that there is more to be said about the relationship between /«(1) and c«(1). In this paper we will look at certain extremal cases of the values of 1«(1) for fixed c«(1), and enumerate those permutations which achieve these extremal values. In a later section we show how our results fit into a broader context of generating the symmetric group by minimal sets of transpositions. We now discuss our notation and terminology. Let[n] = {I, 2, ... , n} and for i,j E [n], i ::::; j, let the interval [i, j] be the set {k E [n] 1 i ::::; k ::::; j}. By Sn we will mean the symmetric group on the set [n]. We will have cause to use various notations for (1 E Sn. If (1 E Sn then (1 is a bijection between [n] and itself. Sometimes we will write (1 = (1, (12 ... (1n where (1i = (1(i) for i E [n] i.e. the second row of the two-line form of (1. Our multiplication of permutations will be from right to left. That is, if (1 = (1'(1\" then (1(i) = (1'«(1\"(i)) for all i E [n]. We willletf«(1) be the number of fixed points in (1, i.e.f«(1) = 1 {i E [n] 1 (1(i) = i} I. Frequently we will write (1 in its cycle decomposition (1 = C, C2 ... Cs where by a cycle C = (c\" C2, ... , cd we mean (1(c;) = CH , for i E [k 1] and (1(Ck) = c,. The standard convention of suppressing the fixed points from the cycle decomposition will be followed. We will abuse notation somewhat and say that a cycle C has a certain property when we really mean that the set of numbers comprising C has that property. Thus, we will say that c is an interval [i, j] if the set {ct 1 Ct E c} = [i, j]. A cycle c = (c\" C2 ... , cd is said to be written in standardform if c, ::::; cj for allj E [k]. Given a subset A of c, A = {Ci' Ci , ... , ci } where i, < i2 < ... < it, the induced cycle I 2 I on A is the cycle (cil , Ci2 ' ... , ci) which may no longer be in standard form. If we delete a set of numbers from a cycle we create the induced cycle on the remaining elements. We will reseve the letter r with various subscripts and superscripts for adjacent transpositions, i.e. r = (i, i + 1) for some i E [n 1], and by ri we mean the particular transposition (i, i + 1). As is well-known, every (1 E Sn can be written as a product of adjacent transpositions. Let /«(1) be the minimum number k of adjacent transpositions such that (1 = r,r2 ••• rk' An inversion in (1 is a pair {(1(i), (1(j)} where (1(i) > (1(j) and i < j. If /«(1) is the set of inversions of (1 then it is also well-known that /«(1) = 1/«(1) I. We will use both these interpretations of /«(1). We will indicate that i andj are in the same cycle of (1 by i ';;' j. If i '\"t' i + 1 and r = ri then we will say that r merges c, which we will notate by rM(1. The significance of this comes from the following elementary theorems which we state without proof.

A characterization of the partial geometry T*-2- (K)

Abstract: In this paper we characterize the partial geometry T2* (K) embedded in AG(3, q) as a net-inducible partial geometry. This characterization is closely related to the characterization theorem of the generalized quadrangle T2*(O) in [8].

Short cocircuits in binary matroids

Abstract: Given a simple graph G having vertex set V, it is obvious that for any spanning tree T, there is an edge of T whose fundamental cutset has size at most |V|— 1. We extend this result to matroids.Call a cocircuit of a matroid M short if its size is at most the rank of M. Then we prove that for any simple binary matroid M having no Fano minor, and for any basis of M, there is a fundamental cocircuit which is short. This theorem gives a simple necessary condition for a matrix to be totally unimodular.

Steiner triple systems with an involution

Abstract: Let π be a permutation of the set {1, 2,..., υ} having f l υ fixed points and (υ — f )/2 disjoint transpositions. We investigate the existence of Steiner triple systems admitting π as an auto-morphism. When f = 1 such a system is known as a reverse Steiner triple system and it is known that reverse Steiner triple systems exist if and only if υ ≡ 1, 3, 9 or 19 (mod 24). In this paper we show that a Steiner triple system admitting π as an automorphism, and f g 1 exists if and only if υ ≡ 1 or 3(mod 6), f ≡ 1 or 3(mod 6), and either (υ — f ≡ 0(mod 4), and υ ⩾ 2 f + 1) or (υ — f ≡ 2 (mod 4), and υ ⩾ 3 f ).

A reciprocity relation for t -designs

Abstract: It is shown that for a given t, ν, κ , the monoid of all t -(ν, κ, λ) designs is a toroidal monoid in the sense of Hochester and Stanley and the corresponding monoid algebra is Gorenstein. Among the consequences deduced is the fact that the function P ν κ, t t(λ) = the number of distinct t -(ν, κ, λ) designs is quasi periodic and satisfies a reciprocity relation. Some generalisations and related problems are discussed.