Defining Sets and Uniqueness in Graph Colorings: a Survey

Perfect $2$-colorings of the Platonic graphs

In this paper, we enumerate the parameter matrices of all perfect $2$-colorings of the Platonic graphs consisting of the tetrahedral graph, the cubical graph, the octahedral graph, the dodecahedral graph, and  the icosahedral graph.

On the Computational Complexity of Defining Sets

Suppose we have a family F of sets. For every S ∈ F , a set D ⊆ S is a defining set for (F , S) if S is the only element of F that contains D as a subset. This concept has been studied in numerous cases, such as vertex colorings, perfect matchings, dominating sets, block designs, geodetics, orientations, and Latin squares. In this paper, first, we propose the concept of a defining set of a logi...

On the Defining Number of (2n - 2)-Vertex Colorings of Kn x Kn

Existence and uniqueness of the solution for a general system of operator equations in $b-$metric spaces endowed with a graph

The purpose of this paper is to present some coupled fixed point results on a metric space endowed with two $b$-metrics. We shall apply a fixed point theorem for an appropriate operator on the Cartesian product of the given spaces endowed with directed graphs. Data dependence, well-posedness and Ulam-Hyers stability are also studied. The results obtained here will be applied to prove the existe...

Complete forcing numbers of polyphenyl systems

The idea of “forcing” has long been used in many research fields, such as colorings, orientations, geodetics and dominating sets in graph theory, as well as Latin squares, block designs and Steiner systems in combinatorics (see [1] and the references therein). Recently, the forcing on perfect matchings has been attracting more researchers attention. A forcing set of M is a subset of M contained...