Journal of Prime Research in Mathematics

Abstract: We use arithmetical tools (algebraic numbers, valuations, Galois groups) in order to describe the structure of all compact subsets of the \(p-\)adic complex field, which are invariant with respect to the absolute Galois group of the \(p-\)adic number field

Abstract: In a given graph \(G = (V;E)\), a set of vertices S with an assignment of colors to them is said to be a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a \(c \geq \chi(G)\) coloring of the vertices of G. A defining set with minimum cardinality is called a minimum defining set and its cardinality is the defining number, denoted by \((d(G; c)\). If F is a family of graphs then \( Spec_{c}(F)=\{d| \exists G, G \epsilon F, d(G,C)=d \} \). Here we study the cases where \(F\) is the family of \(k-\)regular (connected and disconnected) graphs on n vertices and \(c = k-1\). Also the \(Spec_{k-1}(F)\) defining spectrum of all \(k-\)regular (connected and disconnected) graph on n vertices are verified for \(k = 3, 4\) and \(5\).

Abstract: A clique of a graph G is a complete subgraph of \(G\) which is maximal relatively to set inclusion and a covering C of G consisting of s cliques is an irreducible covering if the union of any \(s − 1\) cliques from C is a proper subset of the vertex-set of \(G\). Some discrete optimization problems involve irreducible coverings of graphs: minimization of Boolean functions, minimization of incompletely specified finite automata, finding the chromatic number of a graph. This paper surveys some recent results by the author on the irreducible coverings of graphs by cliques: the recurrence relation and the exponential generating function of the number of irreducible coverings for bipartite graphs, asymptotic behavior of these numbers and of the maximum number of irreducible coverings by cliques of an n-vertex graph as n tends to infinity, extremal graphs of order n for irreducible coverings by \(n − 2\) and \(n − 3\) cliques and the structure of irreducible coverings for bipartite and nonbipartite cases. Some conjectures and open problems are proposed.