χ-bounds, operations, and chords

$\chi$-bounds, operations and chords

A long unichord in a graph is an edge that is the unique chord of some cycle of length at least 5. A graph is long-unichord-free if it does not contain any long-unichord. We prove a structure theorem for long-unichord-free graph. We give an O(nm)-time algorithm to recognize them. We show that any long-unichord-free graph G can be colored with at most O(ω) colors, where ω is the maximum number o...

Upper Bounds for |l(1, Χ)|

Given a non-principal Dirichlet character χ (mod q), an important problem in number theory is to obtain good estimates for the size of L(1, χ). The best bounds known give that q−ǫ ≪ǫ |L(1, χ)| ≪ log q, while assuming the Generalized Riemann Hypothesis, J.E. Littlewood showed that 1/ log log q ≪ |L(1, χ)| ≪ log log q. Littlewood’s result reflects the true range of the size of |L(1, χ)| as it is ...

Effective Lower Bounds for L(1, Χ) via Eisenstein Series

Roughness bounds in rough set operations

This paper presents some roughness bounds for rough set operations. The results show that a bound of the set operation can be determined from their operand’s roughnesses. We prove also that this bound is not determined under some special operations.

Some bounds on convex combinations of ω and χ for decompositions into many parts

Some bounds on convex combinations of ω and χ for decompositions into many parts Abstract A k–decomposition of the complete graph K n is a decomposition of K n into k spanning subgraphs G 1 ,. .. , G k. For a graph parameter p, let p(k; K n) denote the maximum of k j=1 p(G j) over all k–decompositions of K n. It is known that χ(k; K n) = ω(k; K n) for k ≤ 3 and conjectured that this equality ho...