Feasible edge colorings of trees with cardinality constraints

AbstractA variation of preemptive open shop scheduling corresponds to finding a feasible edge coloring in a bipartite multigraph with some requirements on the size of the different color classes. We show that for trees with fixed maximum degree, one can find in polynomial time an edge k-coloring where for i=1,…,k the number of edges of color i is exactly a given number hi, and each edge e gets its color from a set ϕ(e) of feasible colors, if such a coloring exists. This problem is NP-complete for general bipartite multigraphs. Applications to open shop problems with costs for using colors are described

Packing and covering of crossing families of cuts

AbstractLet C be a crossing family of subsets of the finite set V (i.e., if T, U ∈ C and T ⋔ U ≠ ⊘, T ⌣ U ≠ V, then T ⋔ U ∈ C and T ⌣ U ∈ C). If D = (V, A) is a directed graph on V, then a cut induced by C is the set of arcs entering some set in C. A covering for C is a set of arcs entering each set in C, i.e., intersecting all cuts induced by C. It is shown that the following three conditions are equivalent for any given crossing family C: 1.(P1) For every directed graph D = (V, A), the minimum cardinality of a cut induced by C is equal to the maximum number of pairwise disjoint coverings for C.2.(P2) For every directed graph D = (V, A), and for every length function l: A → Z+, the minimum length of a covering for C is equal to the maximum number t of cuts C1,…, Ct induced by C (repetition allowed) such that no arc a is in more than l(a) of these cuts.3.(P3) ⊘ ∈ C, or V ∈ C, or there are no V1, V2, V3, V4, V5 in C such that V1 ⊆ V2 ⋔ V3, V2 ⌣ V3 = V, V3 ⌣ V4 ⊆ V5, V3 ⋔ V4 = ⊘.Directed graphs are allowed to have parallel arcs, so that (P1) is equivalent to its capacity version. (P1) and (P2) assert that certain hypergraphs, as well as their blockers, have the “Z+-max-flow min-cut property”. The equivalence of (P1), (P2), and (P3) implies Menger's theorem, the König-Egerváry theorem, the König-Gupta edge-colouring theorem for bipartite graphs, Fulkerson's optimum branching theorem, Edmonds' disjoint branching theorem, and theorems of Frank, Feofiloff and Younger, and the present author

Some remarks on good colorations

AbstractIn this paper we are concerned with the concept of good k-coloration introduced by Berge. It is shown that if in a hypergraph H no node belongs to more than p edges and if H has a good k-coloration, then there is a good k-coloration S1, S2,…,Sk such that maxi≤k|Si| ≤ (p−1)mini≤k|Si| + 1. Good k-edge colorations of multigraphs are examined and a theorem of Folkman and Fulkerson is extended to the good k-edge colorations and to the equitable k-colorations