#### Date of Award

2014

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematics

#### Department

Mathematics

#### First Advisor

Nancy Eaton

#### Abstract

Several conjectures concerning planar graph colorings are still unsolved to this day. One of the more famous ones is Steinberg's Conjecture (first stated in 1976), which we work towards in this dissertation. Steinberg's Conjecture states that all planar graphs without cycles of length 4 or 5 are 3-colorable, that is, we can color the vertices of such a graph using three colors in a way that leaves no adjacent vertices colored the same.

We use defective colorings to move closer to the affirmation or nullification of this conjecture. A defective coloring is any non-proper coloring, that is, some adjacent vertices may end up colored the same color. We use (d_{1}, d_{2}, d_{3})-colorings, which are 3-colorings where the maximum degree of the *i*^{th} color class is at most *d _{i}*, for

*i*= 1, 2, 3. In this study, we prove that all planar graphs without cycles of length 4 or 5 are (3, 0, 0)-colorable.

We define a cycle having two triangular chords as 2-chorded and a face of size *k* as a *k*-face. Let a b8-face be an 8-face incident to a vertex of degree three, which is itself also incident to faces of size 3 and 6. Let G be the set of all planar graph without C_{4}'s, C_{5}'s, 2-chorded C_{8}'s, or 2-chorded C_{9}'s. In this paper, we prove that all graphs in G with at most eleven b_{8}-faces is (1, 0, 0)-colorable.

We end this dissertation by examining a paper claiming to prove that all planar graphs without 5- and 8-cycles and without adjacent triangles are 3-colorable. We show some counterexamples to a claim given in the paper.

#### Recommended Citation

Smith, Diana L., "TOWARDS STEINBERG'S CONJECTURE" (2014). *Open Access Dissertations.* Paper 212.

https://digitalcommons.uri.edu/oa_diss/212