John R. Schmitt

Warning's Second Theorem with restricted variables

We present a restricted variable generalization of Warning’s Second Theorem (a result giving a lower bound on the number of solutions of a low degree polynomial system over a finite field, assuming one solution exists). This is analogous to Schauz-Brink’s restricted variable generalization of Chevalley’s Theorem (a result giving conditions for a low degree polynomial system not to have exactly one solution). Just as Warning’s Second Theorem implies Chevalley’s Theorem, our result implies Schauz-Brink’s Theorem. We include several combinatorial applications, enough to show that we have a general tool for obtaining quantitative refinements of combinatorial existence theorems.Let q = pℓ be a power of a prime number p, and let Fq be “the” finite field of order q.For a1,...,an, N∈Z+, we denote by m(a1,...,an;N)∈Z+ a certain combinatorial quantity defined and computed in Section 2.1.

On the size and structure of graphs with a constant number of 1-factors

We investigate the maximum number of edges in a graph with a prescribed number of 1-factors. We also examine the structure of such extremal graphs.

A General Lower Bound for Potentially

We consider a variation of the classical Turan-type extremal problem as introduced by Erdos, Jacobson, and Lehel in [Graphs realizing the same degree sequences and their respective clique numbers, in Graph Theory, Combinatorics, and Applications, Vol. 1, Wiley, New York, 1991, pp. 439-449]. Let

H

\pi

-Graphic Sequences

be an

Potentially H-bigraphic sequences

n

On the Alon-Füredi bound

-element graphic sequence and

An Erdős-Stone Type Conjecture for Graphic Sequences

\sigma(\pi)

A Survey of Minimum Saturated Graphs

be the sum of the terms in

Constructive Upper Bounds for Cycle-Saturated Graphs of Minimum Size

\pi