41994930
On the partition and coloring of a graph by cliques
Abstract We first introduce the concept of the k -chromatic index of a graph, and then discuss some of its properties. A characterization of the clique partition number of the graph G ⋁ K c m for any simple graph G is given, together with some of its applications. Graphs with maximum valency 3 are also considered.
38503566
Pseudo-one-factorizations of regular graphs of odd order - II. products of graphs
Abstract We study the pseudo-one-factorization of regular graphs of odd order. It is shown that some classes of products (including composition and cartesian product) of graphs are pseudo-one- factorizable. In particular, we prove that a complete regular t -partite graph of odd order has this property.
36095353
Extremal Regular Graphs with Prescribed Odd Girth
The odd girth of a graph G gives the length of a shortest odd cycle in G. Let ?(k, g) denote the smallest n such that there exists a k-regular graph of order n and odd girth g. It is known that ?(k, g) ? kg/2 and that ?(k, g) = kg/2 if k is even. The exact values of ?(k, g) are also known if k = 3 or g = 5. Let ?x?e denote the smallest even integer no less than x, ?(g) = (?1)g ? 1/2, and s(k) = min {p + q | k = pq, where p and q are both positive integers}. It is proved that if k ? 5 and g ? 7 are both odd, then formula] with the exception that ?(5, 7) = 20.
37771428
Some New Bounds on Single-Change Covering Designs
A single-change covering design SCD(v, k, b) on a c-set V is a sequence of b k-sets (blocks) on V that together cover every pair of elements of V at least once, such that two successive blocks have k-1 elements in common. Let f(v,k) denote the smallest b for which there exists an SCD(v, k, b). Some new upper and lower bounds for f(v,k) are given.
17822576
Resolution in the Smyth powerdomain
Abstract The purposes of this paper are (i) to give a new, efficient representation of Smyth powerdomains as 0-inhabited information systems, following the style of Scott; (ii) to introduce, as a consequence of this representation, a generalized resolution method for an arbitrary Smyth powerdomain and show the soundness and completeness of this method. The connection of (i) with (ii) comes from the observation that the elements of a Smyth powerdomain are semantically equivalent to the basic notion of "clauses' as used in the area of logic programming.
15029835
Quasi-Prime Algebraic Domains
Abstract The purpose of this paper is to introduce a new class of domains called quasi-prime algebraic domains. We show that this class is a good candidate for the purpose of denotational semantics of programming languages. This is achieved by exhibiting many constructions usually needed for denotational semantics on quasi-prime algebraic domains. In the first part we motivate the notion of quasi-primes and quasi-prime algebraic domains by studying the effect of linear functions on domains that are not prime algebraic. In the second part we study categories of quasi-prime algebraic domains by representing them as irreducible information systems. We introduce a symmetric monoidal closed category of quasi-prime algebraic domains with quasi-linear functions. We show further that with the usual Scott continuous functions as morphisms, quasi-prime algebraic domains are cartesian closed. In the third part we show the existence of a saturated quasi-prime algebraic domain, and introduce a framework for solving domain equations in quasi-prime algebraic domains. It is then possible to give, for example, a model for the un-typed lambda calculus with a reflexive quasi-prime algebraic domain.
205118584
dI-Domains as Prime Information Systems
Abstract This paper introduces prime information systems. Prime information systems determine dI-domains and stable approximable mappings give stable functions, and vice versa. The notion of rigid embedding is captured by a subsystem relation. Under this relation, prime information systems form a complete partial order (cpo). Constructions like lifting, sum, product, and function space are proposed which induce continuous functions on the cpo. In this way recursive prime information systems can be defined using fixed point theory.
1726106
Defaults in Domain Theory
Abstract This paper uses ideas from artificial intelligence to show how default notions can be defined over Scott domains. We combine these ideas with ideas arising in domain theory to shed some light on the properties of nonmonotonicity in a general model-theoretic setting. We consider in particular a notion of default nonmonotonic entailment between prime open sets in the Scott topology of a domain. We investigate in what ways this notion obeys the so-called laws of cautious monotony and cautious cut , proposed by Gabbay, Kraus, Lehmann, and Magidor. Our notion of nonmonotonic entailment does not necessarily satisfy cautious monotony, but does satisfy cautious cut. In fact, we show that any reasonable notion of nonmonotonic entailment on prime opens over a Scott domain, satisfying in particular the law of cautious cut, can be concretely represented using our notion of default entailment. We also give a variety of sufficient conditions for defaults to induce cumulative entailments, those satisfying cautious monotony. In particular, we show that defaults with unique extensions are a representation of cumulative nonmonotonic entailment. Furthermore, a simple characterization is given for those default sets which determine unique extensions in coherent domains. Finally, a characterization is given for Scott domains in which default entailment must be cumulative. This is the class of daisy domains ; it is shown to be cartesian closed, a purely domain-theoretic result.
10438151
Complexity of power default reasoning
This paper derives a new and surprisingly low complexity result for inference in a new form of Reiter's propositional default logic (1980). The problem studied here is the default inference problem whose fundamental importance was pointed out by Kraus, Lehmann, and Magidor (1980). We prove that "normal" default inference, in propositional logic, is a problem complete for co-NP(3), the third level of the Boolean hierarchy. Our result (by changing the underlying semantics) contrasts favorably with a similar result of Gottlob (1992), who proves that standard default inference is II/sub 2//sup P/-complete. Our inference relation also obeys all of the laws for preferential consequence relations set forth by Kraus, Lehmann, and Magidor (1990). In particular we get the property of being able to reason by cases and the law of cautious monotony. Both of these laws fail for standard propositional default logic. The key technique for our results is the use of Scott's domain theory to integrate defaults into partial model theory of the logic, instead of keeping defaults as quasiproof rules in the syntax. In particular, reasoning disjunctively entails using the Smyth powerdomain.