1. CMB 2016 (vol 59 pp. 287)
 Dukes, Peter; Lamken, E.R.; Ling, Alan C.H.

An Existence Theory for Incomplete Designs
An incomplete pairwise balanced design is equivalent to a pairwise
balanced design with a distinguished block, viewed as a `hole'.
If there are $v$ points, a hole of size $w$, and all (other)
block sizes equal $k$, this is denoted IPBD$((v;w),k)$. In addition
to congruence restrictions on $v$ and $w$, there is also a necessary
inequality: $v \gt (k1)w$. This article establishes two main existence
results for IPBD$((v;w),k)$: one in which $w$ is fixed and $v$
is large, and the other in the case $v \gt (k1+\epsilon) w$ when
$w$ is large (depending on $\epsilon$). Several possible generalizations
of the problem are also discussed.
Keywords:block design, hypergraph Category:05C70 

2. CMB 2015 (vol 58 pp. 610)
3. CMB 2009 (vol 53 pp. 378)
 Zhou, Sizhong

A New Sufficient Condition for a Graph To Be $(g,f,n)$Critical
Let $G$ be a graph of order $p$, let $a$,
$b$, and $n$ be nonnegative integers with $1\leq a\lt b$, and let $g$
and $f$ be two integervalued functions defined on $V(G)$ such
that $a\leq g(x)\lt f(x)\leq b$ for all $x\in V(G)$. A $(g,f)$factor
of graph $G$ is a spanning subgraph $F$ of $G$ such
that $g(x)\leq d_F(x)\leq f(x)$ for each $x\in V(F)$. Then a graph
$G$ is called $(g,f,n)$critical if after deleting any $n$
vertices of $G$ the remaining graph of $G$ has a $(g,f)$factor.
The binding number $\operatorname{bind}(G)$ of $G$ is the minimum value of
${N_G(X)}/{X}$ taken over all nonempty subsets $X$ of
$V(G)$ such that $N_G(X)\neq V(G)$. In this paper, it is proved
that $G$ is a $(g,f,n)$critical graph if
\[
\operatorname{bind}(G)\gt \frac{(a+b1)(p1)}{(a+1)p(a+b)bn+2}
\quad\text{and}\quad p\geq
\frac{(a+b1)(a+b2)}{a+1}+\frac{bn}{a}.
\]
Furthermore, it is
shown that this
result is best possible in some sense.
Keywords:graph, $(g,f)$factor, $(g,f,n)$critical graph, binding number Category:05C70 

4. CMB 2007 (vol 50 pp. 504)
 Dukes, Peter; Ling, Alan C. H.

Asymptotic Existence of Resolvable Graph Designs
Let $v \ge k \ge 1$ and $\lam \ge 0$ be integers. A \emph{block
design} $\BD(v,k,\lambda)$ is a collection $\cA$ of $k$subsets of a
$v$set $X$ in which every unordered pair of elements from $X$ is
contained in exactly $\lambda$ elements of $\cA$. More generally, for a
fixed simple graph $G$, a \emph{graph design} $\GD(v,G,\lambda)$ is a
collection $\cA$ of graphs isomorphic to $G$ with vertices in $X$ such
that every unordered pair of elements from $X$ is an edge of exactly
$\lambda$ elements of $\cA$. A famous result of Wilson says that for a
fixed $G$ and $\lambda$, there exists a $\GD(v,G,\lambda)$ for all
sufficiently large $v$ satisfying certain necessary conditions. A
block (graph) design as above is \emph{resolvable} if $\cA$ can be
partitioned into partitions of (graphs whose vertex sets partition)
$X$. Lu has shown asymptotic existence in $v$ of resolvable
$\BD(v,k,\lambda)$, yet for over twenty years the analogous problem for
resolvable $\GD(v,G,\lambda)$ has remained open. In this paper, we settle
asymptotic existence of resolvable graph designs.
Keywords:graph decomposition, resolvable designs Categories:05B05, 05C70, 05B10 
