In honour of the work of Alex Rosa (Part II)
Org: Peter Danziger, Tommaso Traetta (Ryerson University)
[PDF]

ANDREA BURGESS, University of New Brunswick
Recent advances on the Hamilton-Waterloo problem  [PDF]

The Hamilton-Waterloo problem, $\mathrm{HWP}(v;m,n;\alpha,\beta)$, asks whether there is a 2-factorization of $K_v$ (for $v$ odd) or $K_v-I$ (for $v$ even) into $\alpha$ $C_m$-factors and $\beta$ $C_n$-factors, where $n \geq m \geq 3$. We present joint work with Peter Danziger and Tommaso Traetta giving a significant advancement towards solving this problem when the cycle lengths $m$ and $n$ are odd. If $v$, $m$, and $n$ are all odd, we show that the obvious necessary conditions are sufficient except possibly if $\beta \in \{1,3\}$, $\alpha=1$, or $v=mn/\gcd(m,n)$. A similar result is obtained for even $v$, with a few further possible exceptional families.

BARBARA MAENHAUT, University of Queensland
Hamilton Decompositions of Line Graphs  [PDF]

Given a graph $G$, the line graph of $G$, denoted $L(G)$, is the graph whose vertices are the edges of $G$ and in which two vertices are adjacent if and only if the corresponding edges of $G$ are adjacent. In this talk we consider conditions on $G$ that ensure that $L(G)$ has a Hamilton decomposition, and present the following theorem: If a $d$-regular graph $G$ is Hamiltonian (for $d$ even) or contains a Hamiltonian 3-factor (for $d$ odd), then $L(G)$ has a Hamilton decomposition.

DAVID PIKE, Memorial University
Colourings of Group Divisible Designs  [PDF]

A group divisible design (GDD) consists of a set $V$, a partition of $V$ into subsets called groups, and a collection ${\cal{B}}$ of subsets of $V$ called blocks such that each 2-subset of $V$ that is a subset of no group is contained within exactly one block. A GDD colouring is a function $f:V\rightarrow{C}$ where $C$ is a set of colours and $|\{f(x):x\in{B}\}|\geqslant{2}$ for each $B\in{\cal{B}}$. The chromatic number of a design is the least cardinality $|C|$ for which a colouring exists. We will discuss recent progress on GDD colourings. This is joint work with Burgess, Danziger, Dinitz and Donovan.

BRETT STEVENS, Carleton University
Kirkman-Hamilton triple systems  [PDF]

A Kirkman-Hamilton triple system is a Hamilton decomposition of $K_n$ where each Hamilton cycle can be cut into $n/3$ paths each containing three vertices and these vertex sets give a Kirkman triple system. We construct Kirkman-Hamilton triple systems for a finite number of value of $n$. This is joint work with Peter Danziger, Nevena Francetic, and Irwin Pressman.

TOMMASO TRAETTA, Ryerson University
Reverse $2$-factorizations via graceful labelings  [PDF]

A $2$-factor of the complete graph is a spanning subgraph whose components are cycles. A $2$-factor is called reverse if it has an involutory automorphism fixing exactly one vertex.

In this talk we show how the concept of a graceful labeling, introduced by Alex Rosa in 1967, can be used along with other tools to construct factorizations of the complete graph into copies of a reverse $2$-factor $F$ provided that a suitable cycle of $F$ is big enough, thus almost completely solving the Oberwolfach problem for reverse $2$-factors. This is joint work with Andrea Burgess and Peter Danziger.