A remark on Hamilton cycles with few colors

Akbari, Etesami, Mahini, and Mahmoody conjectured that every proper edge colouring of Kn with n colours contains a Hamilton cycle with ≤O(logn) colours. They proved that there is always a Hamilton cycle with ≤8n−−√ colours. In this note we improve this bound to O(log3n)

The range of thresholds for diameter 2 in random Cayley graphs

Given a group G, the model G(G,p) denotes the probability space of all Cayley graphs of G where each element of the generating set is chosen independently at random with probability p. Given a family of groups (G_k) and a $c \in \mathbb{R}_+$ we say that c is the threshold for diameter 2 for (G_k) if for any ε > 0 with high probability $\Gamma \in \mathcal{G}(G_k,p)$ has diameter greater than 2 if p \leqslant \sqrt{(c - \eps)\frac{\log{n}}{n}} and diameter at most 2 if p \geqslant \sqrt{(c + \eps)\frac{\log{n}}{n}}. In [5] we proved that if c is a threshold for diameter 2 for a family of groups (G_k) then $c \in [1/4,2]$ and provided two families of groups with thresholds 1/4 and 2 respectively. In this paper we study the question of whether every $c \in [1/4,2]$ is the threshold for diameter 2 for some family of groups. Rather surprisingly it turns out that the answer to this question is negative. We show that every $c \in [1/4,4/3]$ is a threshold but a $c \in (4/3,2]$ is a threshold if and only if it is of the form 4n/(3n-1) for some positive integer n

A remark on Hamilton cycles with few colors

Akbari, Etesami, Mahini, and Mahmoody conjectured that every proper edge colouring of Kn with n colours contains a Hamilton cycle with ≤O(logn) colours. They proved that there is always a Hamilton cycle with ≤8√n colours. In this note we improve this bound to O(log3n)

Partial Latin rectangle graphs and autoparatopism groups of partial Latin rectangles with trivial autotopism groups

An $r \times s$ partial Latin rectangle $(l_{ij})$ is an $r \times s$ matrix containing elements of $\{1,2,\ldots,n\} \cup \{\cdot\}$ such that each row and each column contain at most one copy of any symbol in $\{1,2,\ldots,n\}$. An entry is a triple $(i,j,l_{ij})$ with $l_{ij} \neq \cdot$. Partial Latin rectangles are operated on by permuting the rows, columns, and symbols, and by uniformly permuting the coordinates of the set of entries. The stabilizers under these operations are called the autotopism group and the autoparatopism group, respectively. We develop the theory of symmetries of partial Latin rectangles, introducing the concept of a partial Latin rectangle graph. We give constructions of $m$-entry partial Latin rectangles with trivial autotopism groups for all possible autoparatopism groups (up to isomorphism) when: (a) $r=s=n$, i.e., partial Latin squares, (b) $r=2$ and $s=n$, and (c) $r=2$ and $s \neq n$