A vizing-type theorem for matching forests

A well known Theorem of Vizing states that one can colour the edges of a graph by $\Delta +\alpha$ colours, such that edges of the same colour form a matching. Here, $\Delta$ denotes the maximum degree of a vertex, and $\alpha$ the maximum multiplicity of an edge in the graph. An analogue of this Theorem for directed graphs was proved by Frank. It states that one can colour the arcs of a digraph by $\Delta +\alpha$ colours, such that arcs of the same colour form a branching. For a digraph, $\Delta$ denotes the maximum indegree of a vertex, and $\alpha$ the maximum multiplicity of an arc

Conditions for ß-perfectness

A ß-perfect graph is a simple graph G such that ¿(G') = ß(G') for every induced subgraph G' of G, where ¿(G') is the chromatic number of G', and ß(G') is defined as the maximum over all induced subgraphs H of G' of the minimum vertex degree in H plus 1 (i.e., d(H)+1). The vertices of a ß-perfect graph G can be coloured with ¿(G) colours in polynomial time (greedily). The main purpose of this paper is to give necessary and sufficient conditions, in terms of forbidden induced subgraphs, for a graph to be ß-perfect. We give new sufficient conditions and make improvements to sufficient conditions previously given by others. We also mention a necessary condition which generalizes the fact that no ß-perfect graph contains an even hole

Convex minimization over Z^2

We present an algorithm for minimizing a convex function over all integer vectors in the plane. This problem generalizes both the nearest lattice vector problem and the integer programming problem in the plane

On packing connectors

AbstractGiven an undirected graphG=(V,E) and a partition {S,T} ofV, anS−Tconnector is a set of edgesF⊆Esuch that every component of the subgraph (V,F) intersects bothSandT. We show thatGhaskedge-disjointS-Tconnectors if and only if |δG(V1)∪…∪δG(Vt)|⩾ktfor every collection {V1,…,Vt} of disjoint nonempty subsets ofSand for every such collection of subsets ofT. This is a common generalization of a theorem of Tutte and Nash-Williams on disjoint spanning trees and a theorem of König on disjoint edge covers in a bipartite graph