CanaDAM 2017
Université Ryerson, 12 - 16 juin 2017

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

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.


Atlantic Association for Research in the Mathematical Sciences Centre de recherches mathmatiques The Fields Institute Pacific Institute for the Mathematical Sciences Socit mathmatique du Canada Université Ryerson Office of Naval Research Science and Technology