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The chromatic number of a graph is at most the maximum vertex degree Delta, unless the graph is complete or an odd cycle, in which case Delta+1 colors are required.

A finite, increasing sequence of integers {a_1,...,a_m} such that (a_i-1)|(a_1...a_(m-1)) for i=1, ..., m, where m|n indicates that m divides n. A Carmichael sequence has ...

Defining p_0=2, p_n as the nth odd prime, and the nth prime gap as g_n=p_(n+1)-p_n, then the Cramér-Granville conjecture states that g_n<M(lnp_n)^2 for some constant M>1.

The epispiral is a plane curve with polar equation r=asec(ntheta). There are n sections if n is odd and 2n if n is even. A slightly more symmetric version considers instead ...

An algebraically soluble equation of odd prime degree which is irreducible in the natural field possesses either 1. Only a single real root, or 2. All real roots.

A curvilinear polygon built up of circular arcs. The Reuleaux polygon is a generalization of the Reuleaux triangle and, for an odd number of sides, is a curve of constant ...

Sarnak's constant is the constant C_(Sarnak) = product_(p>=3)(1-(p+2)/(p^3)) (1) = 0.7236484022... (2) (OEIS A065476), where the product is over the odd primes.

Let G be a graph and S a subgraph of G. Let the number of odd components in G-S be denoted S^', and |S| the number of graph vertices of S. The condition |S|>=S^' for every ...

A two-graph (V,Delta) on nodes V is a collection Delta of unordered triples of the vertices (the so-called "odd triples") such that each 4-tuple of V contains an even number ...

An odd prime p is called a cluster prime if every even positive integer less than p-2 can be written as a difference of two primes q-q^', where q,q^'<=p. The first 23 odd ...