creator Mitillos, Christodoulos advisor Kaul, Hemanshu Graph fall-coloring, also known as idomatic partitioning or independent domatic partitioning of graphs, was formally introduced by Dunbar, Hedetniemi, Hedetniemi, Jacobs, Knisely, Laskar, and Rall in 2000 [1] as a simple extension of graph coloring and graph domination. It asks for a partition of the vertex set of a given graph into independent dominating sets. In this thesis, we will study a number of questions related to this concept. In the rst chapter we will give a brief background to graph theory, and introduce the topic of graph fall-coloring, after looking at the fundamental topics it builds on. In the second chapter, we identify the e ects on fall-colorability of various graphical operators, and look at the fall-colorability of certain families of graphs. In the third chapter we will explore certain constructions which create fall-colorable graphs given certain restrictions, and look at the interaction of fall-colorings and non-fall-colorings. Finally, in the fourth chapter, we lay the foundations to establish a connection between fall-coloring and certain existing open problems in graph theory, providing new possible avenues for exploring their solutions. We then provide two applied problems which can be solved with fall-coloring, and which motivate the notion of fall-nearcoloring. We also provide further questions in fall-coloring for future research. Keywords: Graph Fall-coloring, Idomatic Partition, Independent Dominating Sets, Chromatic number, Graph products. Submitted by Erma Thomas (thomase@iit.edu) on 2017-05-19T22:18:39Z No. of bitstreams: 1 etdadmin_upload_450399.zip: 538622 bytes, checksum: baaf1ed5ca8515d5227626dbe61d1e9c (MD5) Made available in DSpace on 2017-05-19T22:18:39Z (GMT). No. of bitstreams: 1 etdadmin_upload_450399.zip: 538622 bytes, checksum: baaf1ed5ca8515d5227626dbe61d1e9c (MD5) Previous issue date: 2016-07 Ph.D. in Applied Mechanics, July 2016 2016 2016-07 http://hdl.handle.net/10560/4081 en Graph Fall-coloring Graph Operators Graph Products Graph Theory Idomatic Partitions Independent Dominating Sets Dissertation born digital application/pdf In Copyright http://rightsstatements.org/page/InC/1.0/ Restricted Access MATH / Applied Mathematics Illinois Institute of Technology Affiliated department