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The distinguishing number of a graph $G$, denoted $D(G)$, is the minimum number of colors needed to produce a coloring of the vertices of $G$ so that every nontrivial isomorphism interchanges vertices of different colors. A list assignment $L$ on a graph $G$ is a function that assigns each vertex of $G$ a set of colors. An $L$-coloring of $G$ is a coloring in which each vertex is colored with a color from $L(v)$. The list distinguishing number of $G$, denoted $D_{ell}(G)$ is the minimum $k$ such that every list assignment $L$ that assigns a list of size at least $k$ to every vertex permits a distinguishing $L$-coloring. In this paper, we prove that when and $n$ is large enough, the distinguishing and list-distinguishing numbers of $K_nBox K_m$ agree for almost all $m>n$, and otherwise differ by at most one. As a part of our proof, we give (to our knowledge) the first application of the Combinatorial Nullstellensatz to the graph distinguishing problem and also prove an inequality for the binomial distribution that may be of independent interest.
A clique minor in a graph G can be thought of as a set of connected subgraphs in G that are pairwise disjoint and pairwise adjacent. The Hadwiger number h(G) is the maximum cardinality of a clique minor in G. This paper studies clique minors in the C
Hellys theorem and its variants show that for a family of convex sets in Euclidean space, local intersection patterns influence global intersection patterns. A classical result of Eckhoff in 1988 provided an optimal fractional Helly theorem for axis-
Given smooth manifolds $M_1,ldots, M_n$ (which may have a boundary or corners), a smooth manifold $N$ modeled on locally convex spaces and $alphain({mathbb N}_0cup{infty})^n$, we consider the set $C^alpha(M_1timescdotstimes M_n,N)$ of all mappings $f
We study several problems concerning convex polygons whose vertices lie in a Cartesian product (for short, grid) of two sets of n real numbers. First, we prove that every such grid contains a convex polygon with $Omega$(log n) vertices and that this
Given a collection of pairwise co-prime integers $% m_{1},ldots ,m_{r}$, greater than $1$, we consider the product $Sigma =Sigma _{m_{1}}times cdots times Sigma _{m_{r}}$, where each $Sigma _{m_{i}}$ is the $m_{i}$-adic solenoid. Answering a question