On the t-Term Rank of a Matrix

For t a positive integer, the t-term rank of a (0,1)-matrix A is defined to be the largest number of 1s in A with at most one 1 in each column and at most t 1s in each row. Thus the 1-term rank is the ordinary term rank. We generalize some basic results for the term rank to the t-term rank, including a formula for the maximum term rank over a nonempty class of (0,1)-matrices with the the same row sum and column sum vectors. We also show the surprising result that in such a class there exists a matrix which realizes all of the maximum terms ranks between 1 and t.Comment: 18 page

Completing partial latin squares with one nonempty row, column, and symbol

Let r,c,s ∈{1,2,…,n} and let PP be a partial latin square of order n in which each nonempty cell lies in row r, column c, or contains symbol s. We show that if n ∉ {3, 4, 5} and row r, column c, and symbol s can be completed in P, then a completion of P exists. As a consequence, this proves a conjecture made by Casselgren and Häggkvist. Furthermore, we show exactly when row r, column c, and symbol s can be completed

Cyclic Matching Sequencibility of Graphs

We define the cyclic matching sequencibility of a graph to be the largest integer $d$ such that there exists a cyclic ordering of its edges so that every $d$ consecutive edges in the cyclic ordering form a matching. We show that the cyclic matching sequencibility of $K_{2m}$ and $K_{2m+1}$ equal $m-1$

On Monochromatic Pairs with Nondecreasing Diameters

Let n m r t be positive integers and Δ : [n] → [r]. We say Δ is (m, r, t) - permissible if there exist t disjoint m-sets B1,…,Bt contained in [n] for which |Δ(Bi)| = 1 for each i = 1,2,…, t. max(Bi) \u3c min(Bi+1) for each i = 1,2,…, t − 1, and max(Bi) − min(Bi) ≤ max(Bi+1) − max(Bi+1) for each i = 1, 2,…, t − 1. Let f(m ,r, t) be the smallest such n so that all colorings Δ are (m, r, t)-permissible. In this paper, we show that f(2, 2, t) = 5t − 4

Circulant Matrices and Mathematical Juggling

Circulants form a well-studied and important class of matrices, and they arise in many algebraic and combinatorial contexts, in particular as multiplication tables of cyclic groups and as special classes of latin squares. There is also a known connection between circulants and mathematical juggling. The purpose of this note is to expound on this connection developing further some of its properties. We also formulate some problems and conjectures with some computational data supporting them