Vitaly I. Voloshin

The Chromatic Spectrum of Mixed Hypergraphs

Abstract. A mixed hypergraph is a triple ℋ=(X, ?, ?), where X is the vertex set, and each of ?, ? is a list of subsets of X. A strict k-coloring of ℋ is a surjection c:X→{1,…,k} such that each member of ? has two vertices assigned a common value and each member of ? has two vertices assigned distinct values. The feasible set of H is {k: H has a strict k-coloring}. Among other results, we prove that a finite set of positive integers is the feasible set of some mixed hypergraph if and only if it omits the number 1 or is an interval starting with 1. For the set {s,t} with 2≤s≤t−2, the smallest realization has 2t−s vertices. When every member of ?∪? is a single interval in an underlying linear order on the vertices, the feasible set is also a single interval of integers.

Uncolorable mixed hypergraphs

Abstract A mixed hypergraph H =(X, A,E ) consists of the vertex set X and two families of subsets: the family E of edges and the family A of co-edges. In a coloring every edge E∈ E has at least two vertices of different colors, while every co-edge A∈ A has at least two vertices of the same color. The largest (smallest) number of colors for which there exists a coloring of a mixed hypergraph H using all the colors is called the upper (lower) chromatic number and is denoted χ ( H ) (χ( H )) . A mixed hypergraph is called uncolorable if it admits no coloring. We show that there exist uncolorable mixed hypergraphs H =(X, A,E ) with arbitrary difference between the upper chromatic number χ ( H A ) of H A =(X, A ) and the lower chromatic number χ( H E ) of H E =(X, E ). Moreover, for any k= χ ( H A )−χ( H E )⩾0 , the minimum number v(k) of vertices of an inclusionwise minimal uncolorable mixed hypergraph is exactly k+4. We introduce a measure of uncolorability (the vertex uncolorability number) and propose a greedy algorithm that finds an estimate on it. We also show that the colorability problem can be expressed in terms of integer programming. Concerning particular cases, we describe those complete (l,m)-uniform mixed hypergraphs which are uncolorable, and observe that for any fixed (l,m) almost all complete (l,m)-uniform mixed hypergraphs are uncolorable, whereas generally almost all complete mixed hypergraphs are colorable.

Colouring Steiner systems with specified block colour patterns

Abstract We consider colourings of Steiner systems S (2,3, v ) and S (2,4, v ) in which blocks have prescribed colour patterns, as a refinement of the classical weak colourings. The main question studied is, given an integer k, does there exist a colouring of given type using exactly k colours? For several types of colourings, a complete answer to this question is obtained while for other types, partial results are presented. We also discuss the question of the existence of uncolourable systems.

Bicolouring Steiner systems S(2,4,v)

We discuss colourings of elements of Steiner systems S(2,4,v) in which the elements of each block get precisely two colours. We show that there exist systems admitting such colourings with arbitrary large number of colours, as well as systems which are uncolourable.

Chromatic spectrum is broken

A mixed hypergraph is a triple H = (X, C,D), where X is the vertex set and each of C, D is a family of subsets of X , the C-edges and D-edges, respectively. A proper k-coloring of a mixed hypergraph is a function from the vertex set to a set of k colors so that each C-edge has two vertices with a common color and each D-edge has two vertices with distinct colors. A mixed hypergraph is k-colorable if it has a proper coloring with at most k colors. A strict k-coloring is a proper k-coloring using all k colors. The minimum number of colors in a strict coloring of H is its lower chromatic number χ(H); the maximum number is its upper chromatic number χ(H).

Problems and Results on Colorings of Mixed Hypergraphs

We survey results and open problems on ‘mixed hypergraphs’ that are hypergraphs with two types of edges. In a proper vertex coloring the edges of the first type must not be monochromatic, while the edges of the second type must not be completely multicolored. Though the first condition just means ‘classical’ hypergraph coloring, its combination with the second one causes rather unusual behavior. For instance, hypergraphs occur that are uncolorable, or that admit colorings with certain numbers k′ and k″ of colors but no colorings with exactly k colors for any k′ < k < k″.

Pseudo-chordal mixed hypergraphs

Abstract A mixed hypergraph contains two families of subsets: edges and co-edges. In every coloring any edge has at least two vertices of different colors, any co-edge has at least two vertices of the same color. The minimum (maximum) number of colors for which there exists a coloring of a mixed hypergraph H using all the colors is called lower (upper) chromatic number. A mixed hypergraph is called uniquely colorable if it has exactly one coloring apart from the permutation of colors. A vertex is called simplicial if its neighborhood induces a uniquely colorable mixed hypergraph. A mixed hypergraph is called pseudo-chordal if it can be decomposed by consecutive elimination of simplicial vertices. The main result of this paper is to provide a necessary and sufficient condition for a polynomial to be a chromatic polynomial of a pseudo-chordal mixed hypergraph.

About uniquely colorable mixed hypertrees

A mixed hypergraph is a triple H = (X, C,D) where X is the vertex set and each of C, D is a family of subsets of X, the C-edges and D-edges, respectively. A k-coloring of H is a mapping c : X → [k] such that each C-edge has two vertices with the same color and each D-edge has two vertices with distinct colors. H = (X, C,D) is called a mixed hypertree if there exists a tree T = (X, E) such that every D-edge and every C-edge induces a subtree of T. A mixed hypergraph H is called uniquely colorable if it has precisely one coloring apart from permutations of colors. We give the characterization of uniquely colorable mixed hypertrees.

Strict colorings of Steiner triple and quadruple systems: a survey

The paper surveys problems, results and methods concerning the coloring of Steiner triple and quadruple systems viewed as mixed hypergraphs. In this setting, two types of conditions are considered: each block of the Steiner system in question has to contain (i) a monochromatic pair of vertices, or, more, restrictively, (ii) a triple of vertices that meets precisely two color classes.

Incidence graphs of biacyclic hypergraphs

Abstract It is well-known that the incidence graphs of totally balanced hypergraphs are exactly chordal bipartite graphs. This paper examines the incidence graphs of biacyclic hypergraphs. We characterize these graphs as absolute bipartite retracts with forbidden isometric wheels, or alternatively via an elimination scheme.