Search results for "Clique number"
showing 2 items of 2 documents
Logical definability of NP-optimisation problems with monadic auxiliary predicates
1993
Given a first-order formula ϕ with predicate symbols e1...el, so,...,sr, an NP-optimisation problem on -structures can be defined as follows: for every -structure G, a sequence of relations on G is a feasible solution iff satisfies ϕ, and the value of such a solution is defined to be ¦S0¦. In a strong sense, every polynomially bounded NP-optimisation problem has such a representation, however, it is shown here that this is no longer true if the predicates s1, ...,sr are restricted to be monadic. The result is proved by an Ehrenfeucht-Fraisse game and remains true in several more general situations.
Bounding the number of vertices in the degree graph of a finite group
2020
Abstract Let G be a finite group, and let cd ( G ) denote the set of degrees of the irreducible complex characters of G . The degree graph Δ ( G ) of G is defined as the simple undirected graph whose vertex set V ( G ) consists of the prime divisors of the numbers in cd ( G ) , two distinct vertices p and q being adjacent if and only if pq divides some number in cd ( G ) . In this note, we provide an upper bound on the size of V ( G ) in terms of the clique number ω ( G ) (i.e., the maximum size of a subset of V ( G ) inducing a complete subgraph) of Δ ( G ) . Namely, we show that | V ( G ) | ≤ max { 2 ω ( G ) + 1 , 3 ω ( G ) − 4 } . Examples are given in order to show that the bound is bes…