David R. Guichard

The number of maximal independent sets in a connected graph

Abstract We determine the maximum number of maximal independent sets which a connected graph on n vertices can have, and we completely characterize the extremal graphs, thereby answering a question of Wilf.

Acyclic graph coloring and the complexity of the star chromatic number

Star chromatic number, introduced by A. Vince, is a natural generalization of chromatic number. We consider the question, “When is χ* < χ?” We show that χ* < χ if and only if a particular digraph is acyclic and that the decisioin problem associated with this question is probably not in NP though it is both NP-hard and NP-easy.

Domination Graphs with Nontrivial Components

Abstract. A tournament is an oriented complete graph. Vertices x and y dominate a tournament T if for all vertices z≠x,y, either (x,z) or (y,z) are arcs in T (possibly both). The domination graph of a tournament T is the graph on the vertex set of T containing edge {x,y} if and only if x and y dominate T. In this paper we determine which graphs containing no isolated vertices are domination graphs of tournaments.

Two theorems on the addition of residue classes

Abstract It is well known that if a 1 ,…, a m are residues modulo n and m ⩾ n then some sum a i 1 + ⋯ + a ik , i 1 i k , is 0 (mod n ). In recent related work, Sydney Bulman-Fleming and Edward T.H. Wang have studied what they call n-divisible subsequences of a finite sequence σ, and made a number of conjectures. We confirm two of those conjectures in a more general form. Let f ( a 1 ,…, a m ; j be the number of sums formed from the a i which are congruent to j (mod n ). We prove two main theorems: 1. If f ( a 1 ,…, a m ; 0) m −1 then f ( a 1 ,hellip;, a m ; 0) ⩽ 3 · 2 m −3 ; 2. Let m ⩾ n . There exist a 1 ,hellip, a m for which f ( a 1 ,hellip:, a m ; j ) is odd if and only if n itis not a power of 2.

Note: Perfect matchings in pruned grid graphs

We determine conditions sufficient to guarantee the existence of a perfect matching when vertices are removed from finite and infinite grid graphs. The conditions impose a minimum distance between the vertices that are removed. While the distances are likely not best possible, they are best possible with respect to asymptotic growth rate.

No-hole k -tuple ( r + 1)-distant colorings of odd cycles

Abstract We give a necessary and sufficient condition for the existence of near-optimal Nkr-colorings of cycles. Troxell (preprint) studied near-optimal Nkr-colorings and proved most of the result presented here; our contribution is to complete the proof for odd cycles.

How Commutative Are Direct Products of Dihedral Groups

Summary If G is a finite group, then Pr(G) is the probability that two randomly selected elements of G commute. So G is abelian iff Pr(G) = 1. For any positive integer m, we show that there is a group G which is a direct product of dihedral groups such that Pr(G) = 1/m. We also show that there is a dihedral group G such that Pr(G) = m/m′, where m′ is relatively prime to m.

When Is U(n) Cyclic? An Algebraic Approach

Early in a typical abstract algebra course we learn that the set U(n) = {O < ? n I gcd(x, n) = 11 is a group under multiplication mod n for eveiy n 2 1. This first appears as example 11 in Chapter 2 of Gallians excellent text [2], for instance. These groups are particularly nice: it is not hard to see, but not immediately obvious, that they are groups; they are important in some modern cryptographic applications; and they figure prominently in elementary number theory. Some of the groups U(n) are cyclic and some are not, and the two categories can be completely characterized by the form of the prime factorization of n1. If U(n) is cyclic then we can write U(n) = K g) for some g E E,/ relatively prime to n. In number theory g is known as a primitive root modulo n; we will call the characterization of those n with primitive roots the Primitive Root Theorem, or PRT. I recently taught an abstract algebra course using Gallians text, and I wanted to prove the PRT for the class. Though this result is standard in elementaiy number theoiy books (see, e.g., [3]), the number-theoretic notation and proofs would have led me farther afield than I cared to go. I failed to find an algebraic proof of the result, but put one together by mining the proof in [3] for hints. The proof uses many results and exercises from [2]; this made it a satisfying conclusion to my course. Most of the proof requires only group theory, though some field theory and experience with polynomial rings is required at the very end.

Statistical Effects of Class Size and Teaching Ability in Determining Teacher Ratings

Abstract Cohen (1983) considered a statistical problem associated with the effect of class size on teacher rating. Specifically, Cohen considered comparisons of mean scores [xbar]A and [xbar]B of two instructors teaching classes of different sizes. A heuristic proof was given that, under certain conditions and among the better lecturers, those who are evaluated by a smaller number of students have an advantage. We examine the effect of class size on teacher ratings for some special cases involving classes of modest size. We discover that class size can have a noticeable impact on teacher ratings but that the advantage lies sometimes in having a larger class and sometimes a smaller. Moreover, it appears that in practice the effect of class size on teacher ratings should be quite small.