Mathematics

A matroid is uniform if and only if it has no minor isomorphic to U 1,1 ??U 0,1 and is paving if and only if it has no minor isomorphic to U 2,2 ??U 0,1 . This paper considers, more generally, when a matroid M has no U k,k ??U 0,??-minor for a fixed pair of positive integers (k,?? . Calling such a matroid (k,?? -uniform, it is shown that this is equivalent to the condition that every rank- (r(M)?�k) flat of M has nullity less than ??. Generalising a result of Rajpal, we prove that for any pair (k,?? of positive integers and prime power q , only finitely many simple cosimple GF(q) -representable matroids are \kl-uniform. Consequently, if Rota's Conjecture holds, then for every prime power q , there exists a pair ( k q , ??q ) of positive integers such that every excluded minor of GF(q) -representability is ( k q , ??q ) -uniform. We also determine all binary (2,2) -uniform matroids and show the maximally 3 -connected members to be Z 5 ?�t,AG(4,2),AG(4,2 ) ??and a particular self-dual matroid P 10 . Combined with results of Acketa and Rajpal, this completes the list of binary (k,?? -uniform matroids for which k+?�≤4 .

The concept of linear set in projective spaces over finite fields was introduced by Lunardon in 1999 and it plays central roles in the study of blocking sets, semifields, rank-distance codes and etc. A linear set with the largest possible cardinality and rank is called maximum scattered. Despite two decades of study, there are only a limited number of maximum scattered linear sets of a line PG(1, q n ) . In this paper, we provide a large family of new maximum scattered linear sets over PG(1, q n ) for any even n?? and odd q . In particular, the relevant family contains at least ????????????????q t +1 8rt ?? ??q t +1 4rt( q 2 +1) ?? if t??2(mod4); if t??(mod4), inequivalent members for given q= p r and n=2t>8 , where p=char( F q ) . This is a great improvement of previous results: for given q and n>8 , the number of inequivalent maximum scattered linear sets of PG(1, q n ) in all classes known so far, is smaller than q 2 . Moreover, we show that there are a large number of new maximum rank-distance codes arising from the constructed linear sets.

We establish the exact lower bound for the number of k -faces of d -polytopes with 2d+1 vertices, for each value of k , and characterise the minimisers. As a byproduct, we characterise all d -polytopes with d+3 vertices, and only one or two edges more than the minimum.

The classical 1961 solution to the problem of determining the number of perfect matchings (or dimer coverings) of a rectangular grid graph -- due independently to Kasteleyn and to Temperley and Fisher -- consists of changing the sign of some of the entries in the adjacency matrix so that the Pfaffian of the new matrix gives the number of perfect matchings, and then evaluating this Pfaffian. Another classical method is to use the Lindström-Gessel-Viennot theorem on non-intersecting lattice paths to express the number of perfect matchings as a determinant, and then evaluate this determinant. In this paper we present a new method for solving the two dimensional dimer problem, which relies on the Cauchy-Binet theorem. It only involves facts that were known in the mid 1930's when the dimer problem was phrased, so it could have been discovered while the dimer problem was still open. We provide explicit product formulas for both the square and the hexagonal lattice. One advantage of our formula for the square lattice compared to the original formula of Kasteleyn, Temperley and Fisher is that ours has a linear number of factors, while the number of factors in the former is quadratic. Our result for the hexagonal lattice yields a formula for the number of periodic stepped surfaces that fit in an infinite tube of given cross-section, which can be regarded as a counterpart of MacMahon's boxed plane partition theorem.

Let Oc t + 1 and Oc t + 2 be the planar and non-planar graphs that obtained from the Octahedron by 3-splitting a vertex respectively. For Oc t + 1 , we prove that a 4-connected graph is Oc t + 1 -free if and only if it is C 2 6 , C 2 2k+1 (k??) or it is obtained from C 2 5 by repeatedly 4-splitting vertices. We also show that a planar graph is Oc t + 1 -free if and only if it is constructed by repeatedly taking 0-, 1-, 2-sums starting from { K 1 , K 2 , K 3 }?�K?�{Oct, L 5 } , where K is the set of graphs obtained by repeatedly taking the special 3-sums of K 4 . For Oc t + 2 , we prove that a 4-connected graph is Oc t + 2 -free if and only if it is planar, C 2 2k+1 (k??) , L( K 3,3 ) or it is obtained from C 2 5 by repeatedly 4-splitting vertices.

In 1988, D. G. Higman gave properties of parameters of strongly regular designs without proof. It included an error. We will give proofs and correct the error.

Recently, Haynes, Hedetniemi and Henning published the book Topics in Domination in Graphs, which comprises 16 contributions that present advanced topics in graph domination, featuring open problems, modern techniques, and recent results. One of these contributions is the chapter Multiple Domination, by Hansberg and Volkmann, where they put into context all relevant research results on multiple domination that have been found up to 2020. In this note, we show how to improve some results on double domination that are included in the book.

We prove a result on the asymptotic proportion of randomly chosen pairs of permutations in the symmetric group S n which "invariably" generate a nonsolvable subgroup, i.e., whose cycle structures cannot possibly both occur in the same solvable subgroup of S n . As an application, we obtain that for a large degree "random" integer polynomial f , reduction modulo two different primes can be expected to suffice to prove the nonsolvability of Gal(f/Q) .

Let p denote a prime number. In this note, we focus on the modular Terwilliger algebras of association schemes defined in [3]. We define the primary module of a modular Terwilliger algebra of an association scheme and determine all its composition factors up to isomorphism. We then characterize the p ??-valenced association schemes by some properties of their modular Terwilliger algebras. The corollaries about the modular Terwilliger algebras of association schemes are given.

Popielarz, Sahasrabudhe and Snyder in 2018 proved that maximal K r+1 -free graphs with (1??1 r ) n 2 2 ?�o( n r+1 r ) edges contain a complete r -partite subgraph on n?�o(n) vertices. This was very recently extended to odd cycles in place of K 3 by Wang, Wang, Yang and Yuan. We further extend it to some other 3-chromatic graphs, and obtain some other stability results along the way.