Color-bounded hypergraphs, VI: Structural and functional jumps in complexity

Abstract

A stably bounded hypergraph H is a hypergraph together with four color-bound functions s, t, a and b, each assigning positive integers to the edges. A vertex coloring of H is considered proper if each edge E has at least s(E) and at most t(E) different colors assigned to its vertices, moreover each color occurs on at most b(E) vertices of E, and there exists a color which is repeated at least a(E) times inside E. The lower and the upper chromatic number of H is the minimum and the maximum possible number of colors, respectively, over all proper colorings. An interval hypergraph is a hypergraph whose vertex set allows a linear ordering such that each edge is a set of consecutive vertices in this order.We study the time complexity of testing colorability and determining the lower and upper chromatic numbers. A complete solution is presented for interval hypergraphs without overlapping edges. Complexity depends both on problem type and on the combination of color-bound functions applied, except that all the three coloring problems are NP-hard for the function pair a,b and its extensions. For the tractable classes, linear-time algorithms are designed. It also depends on problem type and function set whether complexity jumps from polynomial to NP-hard if the instance is allowed to contain overlapping intervals. Comparison is facilitated with three handy tables which also include further structure classes.