For an edge-ordered graph G, an n-vertex edge-ordered graph H is G-saturated if it is G-free and adding any new edge with an arbitrary label to H creates a copy of G. The saturation function is the minimum number of edges in a G-saturated graph. For (unordered) graphs, 0-1 matrices, and vertex-ordered graphs, the saturation function is always O(n) and satisfies a dichotomy: it is either O (1) or Theta(n). The saturation function of an edge-ordered graph follows a weaker dichotomy, being either O (1) or ohm(n). However, by finding edge-ordered graphs whose saturation functions are ohm(n root logn), we show that O(n) is not a universal upper bound. We also study the semisaturation problem for edge-ordered graphs, a variant of the saturation problem in which H is not required to be G-free. We prove a general upper bound O (n log n) and characterize edge-ordered graphs with bounded semisaturation functions. We then present several families of edge-ordered graphs with bounded, linear, and superlinear (semi)saturation functions. We also introduce a natural variant of saturation in which the added edge is required to receive the smallest label. The behaviour of the two variants is similar in many respects, which motivated us to investigate the second variant extensively.
We study the distribution of a partition statistic fsrp(lambda), which is the sum of reciprocals of the different parts of the partition lambda via its moments. The generating functions for the moments are related to polylogarithms, and using Wright's circle method, we derive asymptotic formulas for the moments. In particular, we show that as n-+ oo, root 6n ! ( ( )) E [ fsrp(lambda) : lambda E Pn] = log 1+O n-4 1 pi and (Var [ fsrp(lambda) : lambda E Pn] = O n-4 1 (logn)2), where Pn is the set of of n.
For fixed dd >= 1, let t(n,d) be the number of subsets of [n](d) that tile Z(d) by translations. We prove that t(n,d) = (3(1/3))(nd +/- o(nd).)
A theorem of Li and Ning [Linear Algebra Appl. 515 (2017)] states that, for n 4, every balanced bipartite graph G on 2n vertices with spectral radius lambda(G) n(n - 1) contains a Hamilton path unless G <^> Kn,n-1 U K1. Let [n] = {1, 2, ... , n}. We prove a generalization of this theorem in the setting of graph transversals. Namely, for n 4, we show that given a family g = {G1, G2, ... , G2n-1} of 2n-1 bipartite graphs on a common set V of 2n vertices /with a common balanced bipartition, if lambda(Gi) n(n - 1) for every i E [2n - 1], then there exists a transversal Hamilton path on V unless G1 = G2 = & centerdot; & centerdot; & centerdot; = G2n-1 and G1 <^> Kn,n-1 U K1. We also show spectral analogue of transversal Hamilton cycle in bipartite graph families. Let Bn1 be the graph obtained from Kn,n by deleting all edges in its one subgraph Kn-1,1. Given a family g = {G1, G2,... , G2n} of 2n bipartite graphs on a common set V of 2n vertices with a common balanced bipartition, for n 4, we prove that if lambda(Gi) lambda(Bn1) for every i E [2n], then g contains a transversal Hamilton cycle, unless G1 = G2 = & centerdot; & centerdot; & centerdot; = G2n and G1 <^> Bn1. Our result extends a result of Li and Ning [Linear Multilinear Algebra 64 (11) (2016)].
A Nullstellensatz is a theorem providing information on polynomials that vanish on a certain set: David Hilbert's Nullstellensatz (1893) is a cornerstone of algebraic geometry, and Noga Alon's Combinatorial Nullstellensatz (1999) is a powerful tool in the Polynomial Method, a technique used in combinatorics. Alon's Theorem excludes that a polynomial vanishing on a grid contains a monomial with certain properties. This theorem has been generalized in several directions, two of which we will consider in detail: Terence Tao and Van H. Vu (2006), Uwe Schauz (2008) and Michal Lason (2010) exclude more monomials, and recently, Bogdan Nica (2023) improved the result for grids with additional symmetries in their side edges. Simeon Ball and Oriol Serra (2009) incorporated the multiplicity of zeros and gave Nullstellensa & uml;tze for punctured grids, which are sets of the form X \ Y with both X, Y grids. We generalize some of these results; in particular, we provide a common generalization to the results of Schauz and Nica. To this end, we establish that during multivariate polynomial division, certain monomials are unaffected. This also allows us to generalize Pete L. Clark's proof of Alon and Fu & uml;redi's theorem on the number of nonvanishing points to punctured grids.
