76. CMB 2000 (vol 43 pp. 385)
 Bluskov, I.; Greig, M.; Heinrich, K.

Infinite Classes of Covering Numbers
Let $D$ be a family of $k$subsets (called blocks) of a $v$set
$X(v)$. Then $D$ is a $(v,k,t)$ covering design or covering if every
$t$subset of $X(v)$ is contained in at least one block of $D$. The
number of blocks is the size of the covering, and the minimum size of
the covering is called the covering number. In this paper we consider
the case $t=2$, and find several infinite classes of covering numbers.
We also give upper bounds on other classes of covering numbers.
Categories:05B40, 05D05 

77. CMB 2000 (vol 43 pp. 108)
 Sanders, Daniel P.; Zhao, Yue

On the Entire Coloring Conjecture
The Four Color Theorem says that the faces (or vertices) of a plane
graph may be colored with four colors. Vizing's Theorem says that the
edges of a graph with maximum degree $\Delta$ may be colored with
$\Delta+1$ colors. In 1972, Kronk and Mitchem conjectured that the
vertices, edges, and faces of a plane graph may be simultaneously
colored with $\Delta+4$ colors. In this article, we give a simple
proof that the conjecture is true if $\Delta \geq 6$.
Categories:05C15, 05C10 

78. CMB 2000 (vol 43 pp. 3)
 Adin, Ron; Blanc, David

Resolutions of Associative and Lie Algebras
Certain canonical resolutions are described for free associative and
free Lie algebras in the category of nonassociative algebras. These
resolutions derive in both cases from geometric objects, which in turn
reflect the combinatorics of suitable collections of leaflabeled
trees.
Keywords:resolutions, homology, Lie algebras, associative algebras, nonassociative algebras, Jacobi identity, leaflabeled trees, associahedron Categories:18G10, 05C05, 16S10, 17B01, 17A50, 18G50 

79. CMB 1999 (vol 42 pp. 386)
 Polat, Norbert

Minimal Separators
A separator of a connected graph $G$ is a set of vertices whose
removal disconnects $G$. In this paper we give various conditions
for a separator to contain a minimal one. In particular we prove
that every separator of a connected graph that has no thick end, or
which is of bounded degree, contains a minimal separator.
Category:05C40 

80. CMB 1999 (vol 42 pp. 359)
 Martin, W. J.; Stinson, D. R.

A Generalized Rao Bound for Ordered Orthogonal Arrays and $(t,m,s)$Nets
In this paper, we provide a generalization of the classical Rao
bound for orthogonal arrays, which can be applied to ordered
orthogonal arrays and $(t,m,s)$nets. Application of our new bound
leads to improvements in many parameter situations to the strongest
bounds (\ie, necessary conditions) for existence of these objects.
Categories:05B15, 65C99 

81. CMB 1999 (vol 42 pp. 25)
 Brown, Tom C.; Graham, Ronald L.; Landman, Bruce M.

On the Set of Common Differences in van der Waerden's Theorem on Arithmetic Progressions
Analogues of van der Waerden's theorem on arithmetic progressions
are considered where the family of all arithmetic progressions,
$\AP$, is replaced by some subfamily of $\AP$. Specifically, we
want to know for which sets $A$, of positive integers, the
following statement holds: for all positive integers $r$ and $k$,
there exists a positive integer $n= w'(k,r)$ such that for every
$r$coloring of $[1,n]$ there exists a monochromatic $k$term
arithmetic progression whose common difference belongs to $A$. We
will call any subset of the positive integers that has the above
property {\em large}. A set having this property for a specific
fixed $r$ will be called {\em $r$large}. We give some necessary
conditions for a set to be large, including the fact that every
large set must contain an infinite number of multiples of each
positive integer. Also, no large set $\{a_{n}: n=1,2,\dots\}$ can
have $\liminf\limits_{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}} > 1$.
Sufficient conditions for a set to be large are also given. We
show that any set containing $n$cubes for arbitrarily large $n$,
is a large set. Results involving the connection between the
notions of ``large'' and ``2large'' are given. Several open
questions and a conjecture are presented.
Categories:11B25, 05D10 

82. CMB 1998 (vol 41 pp. 33)
 Gallant, Robert; Colbourn, Charles J.

Asymptotic existence of tight orthogonal main effect plans
Our main result is showing the asymptotic existence of tight
$\OMEP$s. More precisely, for each fixed number $k$ of rows, and with the
exception of $\OMEP$s of the form $2 \times 2 \times \cdots 2 \times 2s\specdiv 4s$
with $s$ odd and with more than three rows, there are only a finite number
of tight $\OMEP$ parameters for which the tight $\OMEP$ does not exist.
Categories:62K99, 05B15 
