Upper Bounds on Sets of Orthogonal Colorings of Graphs

We generalize the notion of orthogonal latin squares to colorings of simple graphs. Two $n$-colorings of a graph are said to be \emph{orthogonal} if whenever two vertices share a color in one coloring they have distinct colors in the other coloring. We show that the usual bounds on the maximum size of a certain set of orthogonal latin structures such as latin squares, row latin squares, equi-$n$ squares, single diagonal latin squares, double diagonal latin squares, or sudoku squares are a special cases of bounds on orthogonal colorings of graphs.Comment: 17 page

On growth functions of ordered hypergraphs

For $k,l\ge2$ we consider ideals of edge $l$-colored complete $k$-uniform hypergraphs $(n,\chi)$ with vertex sets $[n]=\{1, 2, \dots n\}$ for $n\in\mathbb{N}$. An ideal is a set of such colored hypergraphs that is closed to the relation of induced ordered subhypergraph. We obtain analogues of two results of Klazar [arXiv:0703047] who considered graphs, namely we prove two dichotomies for growth functions of such ideals of colored hypergraphs. The first dichotomy is for any $k,l\ge2$ and says that the growth function is either eventually constant or at least $n-k+2$. The second dichotomy is only for $k=3,l=2$ and says that the growth function of an ideal of edge two-colored complete $3$-uniform hypergraphs grows either at most polynomially, or for $n\ge23$ at least as $G_n$ where $G_n$ is the sequence defined by $G_1=G_2=1$, $G_3=2$ and $G_n = G_{n-1} + G_{n-3}$ for $n\ge4$. The lower bounds in both dichotomies are tight.Comment: 43 pages, 7 figure

On the 1-switch conjecture in the Hypercube and other graphs

Feder and Subi conjectured that for any $2$-coloring of the edges of the $n$-dimensional cube, we can find an antipodal pair of vertices connected by a path that changes color at most once. We discuss the case of random colorings, and we prove the conjecture for a wide class of colorings. Our method can be applied to a more general problem, where $Q_n$ can be replaced by any graph $G$, the notion of antipodality by a fixed automorphism $\phi \in Aut(G)$. Thus for any $2$-coloring of $E(G)$ we are looking for a pair of vertices $u,v$ such that $u= \phi(v)$ and there is a path between them with as few color changes as possible. We solve this problem for the toroidal grid $G=C_{2a} \square c_{2b}$ with the automorphism that takes every vertex to its unique farthest pair. Our results point towards a more general conjecture which turns out to be supported by a previous theorem of Feder and Subi.Comment: 11 pages, 2 figure

Maximal ambiguously k-colorable graphs

A graph is ambiguously k-colorable if its vertex set admits two distinct partitions each into at most k anticliques. We give a full characterization of the maximally ambiguously k-colorable graphs in terms of quadratic matrices. As an application, we calculate the maximum number of edges an ambiguously k-colorable graph can have, and characterize the extremal graphs

New Lower Bounds for 28 Classical Ramsey Numbers

We establish new lower bounds for $28$ classical two and three color Ramsey numbers, and describe the heuristic search procedures we used. Several of the new three color bounds are derived from the two color constructions; specifically, we were able to use $(5,k)$-colorings to obtain new $(3,3,k)$-colorings, and $(7,k)$-colorings to obtain new $(3,4,k)$-colorings. Some of the other new constructions in the paper are derived from two well-known colorings: the Paley coloring of $K_{101}$ and the cubic coloring of $K_{127}$

On Pyber's base size conjecture

Let $G$ be a permutation group on a finite set $\Omega$. A subset $B \subseteq \Omega$ is a base for $G$ if the pointwise stabilizer of $B$ in $G$ is trivial. The base size of $G$, denoted $b(G)$, is the smallest size of a base. A well known conjecture of Pyber from the early 1990s asserts that there exists an absolute constant $c$ such that $b(G) \le c\log |G| / \log n$ for any primitive permutation group $G$ of degree $n$. Some special cases have been verified in recent years, including the almost simple and diagonal cases. In this paper, we prove Pyber's conjecture for all non-affine primitive groups.Comment: 18 pages; to appear in Trans. Amer. Math. So

A note on Dehn colorings and invariant factors

If $A$ is an abelian group and $\phi$ is an integer, let $A(\phi)$ be the subgroup of $A$ consisting of elements $a \in A$ such that $\phi \cdot a=0$. We prove that if $D$ is a diagram of a classical link $L$ and $0=\phi_0,\phi_1,\dots,\phi_{n-1}$ are the invariant factors of an adjusted Goeritz matrix of $D$, then the group $\mathcal{D}_{A}(D)$ of Dehn colorings of $D$ with values in $A$ is isomorphic to the direct product of $A$ and $A=A(\phi_{0}),A(\phi_1),\dots,A(\phi_{n-1})$. It follows that the Dehn coloring groups of $L$ are isomorphic to those of a connected sum of torus links $T_{(2,\phi_1)} \text{ }\# \text{ } \cdots \text{ } \# \text{ } T_{(2,\phi_{n-1})}$.Comment: v1: 10 pages, 4 figures. v2: 9 pages, 3 figures. Further changes may be made before publication in the Journal of Knot Theory and its Ramification

Minimum Number of Fox Colors for Small Primes

This article concerns exact results on the minimum number of colors of a Fox coloring over the integers modulo r, of a link with non-null determinant. Specifically, we prove that whenever the least prime divisor of the determinant of such a link and the modulus r is 2, 3, 5, or 7, then the minimum number of colors is 2, 3, 4, or 4 (respectively) and conversely. We are thus led to conjecture that for each prime p there exists a unique positive integer, m, with the following property. For any link L of non-null determinant and any modulus r such that p is the least prime divisor of the determinant of L and the modulus r, the minimum number of colors of L modulo r is m.Comment: 12 pages, 2 figures, version accepted in JKT

Some properties of Bowlin and Brin's color graphs

Bowlin and Brin defined the class of color graphs, whose vertices are triangulated polygons compatible with a fixed four-coloring of the polygon vertices. In this article it is proven that each color graph has a vertex-induced embedding in a hypercube, and an upper bound is given for the hypercube dimension. The color graphs for $n$-gons up to $n=8$ are listed and some of their features are discussed. Finally it is shown that certain color graphs cannot be isometrically embedded in a hypercube of any dimension.Comment: 24 pages, many figures. One minor rephrasing, and support footnote moved to an Acknowledgements sectio

Computational lower limits on small Ramsey numbers

Computer-based attempts to construct lower bounds for small Ramsey numbers are discussed. A systematic review of cyclic Ramsey graphs is attempted. Many known lower bounds are reproduced. Several new bounds are reported