Every integer is either even or odd, so we know that the polynomial f(x) = x(x− 1) 2 is integervalued on the integers, even though its coefficients are not in Z. Similarly, since every binomial coefficient ( k n ) is an integer, the polynomial ( x n ) = x(x− 1)...(x− n+ 1) n! must also be integervalued. These polynomials were used for polynomial interpolation as far back as the 17 century. Inte...

Stochastic Boolean satisfiability (SSAT) is a formalism allowing decision-making for optimization under quantitative constraints. Although SSAT solvers are active development, existing do not provide Skolem-function witnesses, which crucial practical applications. In this work, we develop new witness-generating solver, SharpSSAT, integrates techniques, including component caching, clause learni...

In 1954, Tutte conjectured that every bridgeless graph has a nowherezero 5-flow. Let ω be the minimum number of odd cycles in a 2-factor of a bridgeless cubic graph. Tutte’s conjecture is equivalent to its restriction to cubic graphs with ω ≥ 2. We show that if a cubic graph G has no edge cut with fewer than 5 2ω − 1 edges that separates two odd cycles of a minimum 2-factor of G, then G has a n...

Let F be a finite field of odd cardinality. A polynomial g in F[x] is called a permutation polynomial if g defines a bijective function on F. We will call a polynomial f in F[x] a difference permutation polynomial if f(x + a) -f(x) is a permutation polynomial for every nonzero a in F. Difference permutation polynomials are a special case of planar functions, as defined by Dembowski [2, p. 2271,...

A long-standing conjecture asserts the existence of a positive constant c such that every simple graph of order n without isolated vertices contains an induced subgraph of order at least cn such that all degrees in this induced subgraph are odd. Radcliffe and Scott have proved the conjecture for trees, essentially with the constant c = 2/3. Scott proved a bound for c depending on the chromatic ...

An odd hole is an induced odd cycle of length at least 5. Scott and Seymour confirmed a conjecture of Gyárfás and proved that if a graph G has no odd holes then χ(G) 6 22 ω(G)+2 . Chudnovsky, Robertson, Seymour and Thomas showed that if G has neither K4 nor odd holes then χ(G) 6 4. In this note, we show that if a graph G has neither triangles nor quadrilaterals, and has no odd holes of length a...

A longstanding open problem in lambda calculus is whether there exists a non-syntactical model of the untyped lambda calculus whose theory is exactly the least λ-theory λβ. In this paper we make use of the Visser topology for investigating the related question of whether the equational theory of a model can be recursively enumerable (r.e. for brevity). We introduce the notion of an effective mo...

A shell graph is the join of a path P k of 'k' vertices and K 1. A subdivided shell graph can be constructed by subdividing the edges in the path of the shell graph. In this paper we prove that the disjoint union of two subdivided shell graphs is odd graceful and also one modulo three graceful. 1. Introduction A graph labeling is an assignment of integers to the vertices or edges, or both, subj...

A binary clutter is cycling if its packing and covering linear program have integral optimal solutions for all Eulerian edge capacities. We prove that the clutter of odd stwalks of a signed graph is cycling if and only if it does not contain as a minor the clutter of odd circuits of K5 nor the clutter of lines of the Fano matroid. Corollaries of this result include, of many, the characterizatio...