Defining sets in vertex colorings of graphs and latin rectangles

Defining Sets and Uniqueness in Graph Colorings: a Survey

There are different ways of coloring a graph, namely vertex coloring, edge coloring, total coloring, list coloring, etc. Literature is full of fascinating papers and even books and monographs on this subject. This note will briefly survey some results on two concepts in graph colorings. First we discuss the uniqueness of graph colorings and then we introduce the concept of defining sets on this...

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

On Uniquely List Colorable Graphs

Let G be a graph with n vertices and suppose that for each vertex v in G, there exists a list of k colors, L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k–list colorable graph. Recently M. Mahdian and E.S. Mahmoodian characterized uniquely 2–list colorable graphs. Here we state some results which will pave the way in character...

J un 1 99 9 On Uniquely List Colorable Graphs ∗

Let G be a graph with n vertices and suppose that for each vertex v in G, there exists a list of k colors, L(v), such that there is a unique proper coloring for G from this collection of lists, then G is called a uniquely k–list colorable graph. Recently M. Mahdian and E.S. Mahmoodian characterized uniquely 2–list colorable graphs. Here we state some results which will pave the way in character...