Kernels by Monochromatic Directed Paths in 3-Colored Tournaments and Quasi-Tournaments
We call the tournament T an m-coloured tournament if the arcs of T are coloured with m colours. If v is a vertex of an m-coloured tournament T, we denote by ζ(v) the set of colours assigned to the arcs with v as endpoint, ζ − (v) the set of colours assigned to the arcs with v as final vertex, and ζ + (v) the set of colours assigned to the arcs with v as an initial vertex. In this paper, we prove that if T is a 3-coloured tournament which does not contain C3 such that: 1. for each v ∈ V(T) |ζ − (v) |≤ 2,
On monochromatic paths and monochromatic cycles in edge coloured tournaments
Abstract We call the tournament T an m-coloured tournament if the arcs of T are coloured with m colours. A directed cycle is called a quasi-monochromatic cycle if with at most one exception all of its arcs are coloured alike. In this paper I obtain some sufficient conditions for an m -coloured tournament T to have a vertex v such that for every other vertex x of T there is a monochromatic directed path from x to v . In particular I prove: If T is an m -coloured tournament such that every directed cycle of length at most 4 is a quasi-monochromatic cycle then there is a vertex v of T such that for every other vertex x of T there is a monochromatic directed path from x to v .
Representations and computations of {2, 3~} and {2, 4~}-inverses in indefinite inner product spaces
We investigate full-rank representations of { 2 , 3 ~ } and { 2 , 4 ~ } -inverses with prescribed rank as well as with prescribed range and null space in an indefinite inner product space. In certain particular cases we obtain new results or generalizations related with { 2 , 3 } and { 2 , 4 } -inverses in a Hilbert (unitary) space. Direct and iterative methods for computing { 2 , 3 ~ } and { 2 , 4 ~ } -inverses in an indefinite inner product space are developed. Numerical examples with different metric matrices are presented.