On constants in the Füredi–Hajnal and the Stanley–Wilf conjecture

AbstractFor a given permutation matrix P, let fP(n) be the maximum number of 1-entries in an n×n (0,1)-matrix avoiding P and let SP(n) be the set of all n×n permutation matrices avoiding P. The Füredi–Hajnal conjecture asserts that cP:=limn→∞fP(n)/n is finite, while the Stanley–Wilf conjecture asserts that sP:=limn→∞|SP(n)|n is finite.In 2004, Marcus and Tardos proved the Füredi–Hajnal conjecture, which together with the reduction introduced by Klazar in 2000 proves the Stanley–Wilf conjecture.We focus on the values of the Stanley–Wilf limit (sP) and the Füredi–Hajnal limit (cP). We improve the reduction and obtain sP⩽2.88cP2 which decreases the general upper bound on sP from sP⩽constconstO(klog(k)) to sP⩽constO(klog(k)) for any k×k permutation matrix P. In the opposite direction, we show cP=O(sP4.5).For a lower bound, we present for each k a k×k permutation matrix satisfying cP=Ω(k2)

A result on polynomials derived via graph theory

We present an example of a result in graph theory that is used to obtain a result in another branch of mathematics. More precisely, we show that the isomorphism of certain directed graphs implies that some trinomials over finite fields have the same number of roots

Degrees of nonlinearity in forbidden 0–1 matrix problems

AbstractA 0–1 matrix A is said to avoid a forbidden 0–1 matrix (or pattern) P if no submatrix of A matches P, where a 0 in P matches either 0 or 1 in A. The theory of forbidden matrices subsumes many extremal problems in combinatorics and graph theory such as bounding the length of Davenport–Schinzel sequences and their generalizations, Stanley and Wilf’s permutation avoidance problem, and Turán-type subgraph avoidance problems. In addition, forbidden matrix theory has proved to be a powerful tool in discrete geometry and the analysis of both geometric and non-geometric algorithms.Clearly a 0–1 matrix can be interpreted as the incidence matrix of a bipartite graph in which vertices on each side of the partition are ordered. Füredi and Hajnal conjectured that if P corresponds to an acyclic graph then the maximum weight (number of 1s) in an n×n matrix avoiding P is O(nlogn). In the first part of the article we refute of this conjecture. We exhibit n×n matrices with weight Θ(nlognloglogn) that avoid a relatively small acyclic matrix. The matrices are constructed via two complementary composition operations for 0–1 matrices. In the second part of the article we simplify one aspect of Keszegh and Geneson’s proof that there are infinitely many minimal nonlinear forbidden 0–1 matrices. In the last part of the article we investigate the relationship between 0–1 matrices and generalized Davenport–Schinzel sequences. We prove that all forbidden subsequences formed by concatenating two permutations have a linear extremal function

On Murty-Simon Conjecture II

A graph is diameter two edge-critical if its diameter is two and the deletion of any edge increases the diameter. Murty and Simon conjectured that the number of edges in a diameter two edge-critical graph on $n$ vertices is at most $\lfloor \frac{n^{2}}{4} \rfloor$ and the extremal graph is the complete bipartite graph $K_{\lfloor \frac{n}{2} \rfloor, \lceil \frac{n}{2} \rceil}$. In the series papers [7-9], the Murty-Simon Conjecture stated by Haynes et al. is not the original conjecture, indeed, it is only for the diameter two edge-critical graphs of even order. In this paper, we completely prove the Murty-Simon Conjecture for the graphs whose complements have vertex connectivity $\ell$, where $\ell = 1, 2, 3$; and for the graphs whose complements have an independent vertex cut of cardinality at least three.Comment: 9 pages, submitted for publication on May 10, 201