#### Degree

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Document Type

Dissertation

#### Abstract

This dissertation consists of two parts. In the first part, I examine Seymour’s Second-Neighborhood Conjecture, which states that every orientation of every simple graph has at least one vertex v such that the number of vertices of out-distance 2 from v is at least as large as the number of vertices of out-distance 1 from it. I present alternative statements of this conjecture using the language of linear algebra, the last one being completely in terms of the inverse of some matrix. In the second part of this dissertation, comprising of Chapters 2 and 3, I examine two conjectures on graph decompositions. The first one proposes that every even order hypercube Q2n has a symmetric Hamilton decomposition, meaning that every cycle can be derived from every other cycle just by permuting the axes. I show that this conjecture holds when n is of the form 2^a.3^b. The second conjecture states that for every graph G its edge set can be partitioned into two sets E1 and E2 such that the contractions G/E1 and G/E2 are K4-minor free. This conjecture is currently open, but I ask and answer two slightly different questions: If I use three sets in the partition, contracting two sets at a time, I can avoid K4 as a minor, but if I use two sets in the partition, contracting one set at a time, there are some graphs that force a K2,3 minor.

#### Recommended Citation

Bouya, Farid, "Some Results on Seymour’s Second-Neighborhood Conjecture and on Decompositions of Graphs" (2020). *LSU Doctoral Dissertations*. 5327.

https://digitalcommons.lsu.edu/gradschool_dissertations/5327

#### Committee Chair

Oporowski, Bogdan