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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)

 Resolutions of Associative and Lie Algebras Certain canonical resolutions are described for free associative and free Lie algebras in the category of non-associative algebras. These resolutions derive in both cases from geometric objects, which in turn reflect the combinatorics of suitable collections of leaf-labeled trees. Keywords:resolutions, homology, Lie algebras, associative algebras, non-associative algebras, Jacobi identity, leaf-labeled trees, associahedronCategories: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 2-large'' 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
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