Pseudo-chordal mixed hypergraphs

AbstractA 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

Weak Colorings of Computable Hypergraphs

After introducing the reader to hypergraphs and their colorings, we generalize computable and highly computable graphs to develop the notion of computable and highly computable hypergraphs. If for a graph G we define x(G) as the chromatic number of G and xC(G)to be the computable chromatic number of G, then Bean showed that for every connected computable and highly computable graph G where x(G) = 2, then xC(G) = 2. We show that there exists a 3-uniform, connected hypergraph H such that xH) = 2 and xC(H) = 1. Furthermore, we show that there exists a connected highly computable hypergraph H such that x(H) = 2 and xC(H) = 3. Lastly, we show that for every highly computable hypergraph H where x(H) = k, it follows that xC(H) x 2k

Mixed interval hypergraphs

AbstractWe investigate the coloring properties of mixed interval hypergraphs having two families of subsets: the edges and the co-edges. In every edge at least two vertices have different colors. The notion of a co-edge was introduced recently in Voloshin (1993, 1995): in every such a subset at least two vertices have the same color. The upper (lower) chromatic number is defined as a maximum (minimum) number of colors for which there exists a coloring of a mixed hypergraph using all the colors.We find that for colorable mixed interval hypergraph H the lower chromatic number χ(H) ⩽ 2, the upper chromatic number χ(H) = |X|−s(H), where s(H) is introduced as the so-called sieve number. A characterization of uncolorability of a mixed interval hypergraph is found, namely: such a hypergraph is uncolorable if and only if it contains an obviously uncolorable edge.The co-stability number α.√(H) is the maximum cardinality of a subset of vertices which contains no co-edge. A mixed hypergraph H is called co-perfect if χ(H′) = α√(H′) for every subhypergraph H′. Such minimal non-co-perfect hypergraphs as monostars and cycloids Cr2r−1 have been found in Voloshin (1995). A new class of non-co-perfect mixed hypergraphs called covered co-bi-stars is found in this paper. It is shown that mixed interval hypergraphs are coperfect if and only if they do not contain co-monostars and covered co-bi-stars as subhypergraphs.Linear time algorithms for computing lower and upper chromatic numbers and respective colorings for this class of hypergraphs are suggested

Chromatic Polynomials of Some Mixed Hypergraphs

Motivated by a recent result of M. Walter [Electron. J. Comb. 16, No. 1, Research Paper R94, 16 p. (2009; Zbl 1186.05059)] concerning the chromatic polynomials of some hypergraphs, we present the chromatic polynomials of several (non-uniform) mixed hypergraphs. We use a recursive process for generating explicit formulae for linear mixed hypercacti and multi-bridge mixed hypergraphs using a decomposition of the underlying hypergraph into blocks, defined via chains. Further, using an algebra software package such as Maple, one can use the basic formulae and process demonstrated in this paper to generate the chromatic polynomials for any linear mixed hypercycle, unicyclic mixed hypercyle, mixed hypercactus and multi-bridge mixed hypergraph. We also give the chromatic polynomials of several examples in illustration of the process including the formulae for some mixed sunflowers

About perfection of circular mixed hypergraphs

A mixed hypergraph is a triple H = (X,C,D), where X is the vertex set and each of C and D is a family of subsets of X, the C-edges and D-edges, respectively. A proper k-coloring of H is a mapping c : X → {1,...,k} such that each C-edge has two vertices with a common color and each D-edge has two vertices with different colors. Maximum number of colors in a coloring using all the colors is called upper chromatic number χ ̄(H). Maximum cardinality of subset of vertices which contains no C-edge is C-stability number αC (H). A mixed hypergraph is called C-perfect if χ ̄ (H') = αC (H') for any induced subhypergraph H'. A mixed hyper- graph H is called circular if there exists a host cycle on the vertex set X such that every edge (C- or D-) induces a connected subgraph on the host cycle. We give a characterization of C-perfect circular mixed hypergraphs

Gr\"obner Bases and Nullstellens\"atze for Graph-Coloring Ideals

We revisit a well-known family of polynomial ideals encoding the problem of graph-$k$-colorability. Our paper describes how the inherent combinatorial structure of the ideals implies several interesting algebraic properties. Specifically, we provide lower bounds on the difficulty of computing Gr\"obner bases and Nullstellensatz certificates for the coloring ideals of general graphs. For chordal graphs, however, we explicitly describe a Gr\"obner basis for the coloring ideal, and provide a polynomial-time algorithm.Comment: 16 page

Műszaki informatikai problémákhoz kapcsolódó diszkrét matematikai modellek vizsgálata = Discrete mathematical models related to problems in informatics

Diszkrét matematikai módszerekkel strukturális és kvantitatív összefüggéseket bizonyítottunk; algoritmusokat terveztünk, komplexitásukat elemeztük. Az eredmények a gráfok és hipergráfok elméletéhez, valamint on-line ütemezéshez kapcsolódnak. Néhány kiemelés: - Pontosan leírtuk azokat a szerkezeti feltételeket, amelyeknek teljesülni kell ahhoz, hogy egy kommunikációs hálózatban és annak minden összefüggő részében legyen olyan, megadott típusú összefüggő részhálózat, ahonnan az összes többi elem közvetlenül elérhető. (A probléma két évtizeden át megoldatlan volt.) - Aszimptotikusan pontos becslést adtunk egy n-elemű alaphalmaz olyan, k-asokból álló halmazrendszereinek minimális méretére, amelyekben minden k-osztályú partícióhoz van olyan halmaz, ami az összes partíció-osztályt metszi. (Nyitott probléma volt 1973 óta, több szerző egymástól függetlenül is felvetette.) - Halmazrendszerek partícióira az eddigieknél általánosabb modellt vezettünk be, megvizsgáltuk részosztályainak hierarchikus szerkezetét és hatékony algoritmusokat adtunk. (Sok alkalmazás várható az erőforrás-allokáció területén.) - Kidolgoztunk egy módszert, amellyel lokálisan véges pozíciós játékok nyerő stratégiája megtalálható mindössze lineáris méretű memória használatával. - Félig on-line ütemezési algoritmusokat terveztünk (kétgépes feladatra, nem azonos sebességű processzorokra), amelyeknek versenyképességi aránya bizonyítottan jobb, mint ami a legjobb teljesen on-line módszerekkel elérhető. | Applying discrete mathematical methods, we proved structural and quantitative relations, designed algorithms and analyzed their complexity. The results deal with graph and hypergraph theory and on-line scheduling. Some selected ones are: - We described the exact structural conditions which have to hold in order that an intercommunication network and each of its connected parts contain a connected subnetwork of prescribed type, from which all the other nodes of the network can be reached via direct link. (This problem was open for two decades.) - We gave asymptotically tight estimates on the minimum size of set systems of k-element sets over an n-element set such that, for each k-partition of the set, the set system contains a k-set meeting all classes of the partition. (This was an open problem since 1973, raised by several authors independently.) - We introduced a new model, more general than the previous ones, for partitions of set systems. We studied the hierarchic structure of its subclasses, and designed efficient algorithms. (Many applications are expected in the area of resource allocation.) - We developed a method to learn winning strategies in locally finite positional games, which requires linear-size memory only. - We designed semi-online scheduling algorithms (for two uniform processors of unequal speed), whose competitive ratio provably beats the best possible one achievable in the purely on-line setting