# User:Maxblack45

## Contents

- 1
**Project Description** - 2
**Background Readings** - 3
**Artifacts** - 4
**Project Log**

**Project Description**

This project focuses on expanding the theoretical applications of *(i,j)-step competition graphs*. Specifically, this project explores an extension of the (*i,j*)-step competition graph from digraphs to directed hypergraphs (hyperdigraphs)--objects whose arcs consist of two disjoint (and possibly empty) sets, the tail and head set (see figure). After establishing a definition for our extension, we will investigate its innate attributes concerning its construction and applications. As the project progresses, results are finalized, and the paper is ready for submission/publication, more details will be added.

My weekly log can be found below and after it, you may find my artifacts from the summer.

**Background Readings**

I felt having a list of the background reading I am doing would be helpful. Hence, I have created one below. These readings include the ones Dr. Factor gave me and the ones I have found. Note: if a link is missing, the reading is either from a textbook or I was unable to find one.

*BioMath: The Biology and Mathematics of Food Webs*by Midge Cozzens, Nancy Crisler, Randi Rotjan, and Tom Fleetwood- Chapter 3 from
*Applied Combinatorics*by Fred S. Roberts *(i,j) Competition Graphs*by Kim A.S. Hefner, Kathryn F. Jones, Suh-ryung Kim, J. Richard Lundgren, and Fred S. Roberts.*The (1,2)-step competition graph of a tournament*by Kim A.S. Factor and Sarah K. Merz*An Introduction to (1,2)-Domination Graphs*by Kim A.S. Factor*The m-step, same-step, and any-step competition graphs*by Wei ho*Connected triangle-free m-step competition graphs*by Geir T. Helleloid*The m-step competition graph of a digraph*by Han Hyuk Cho, Suh-Ryun Kim, and Yunsun Nam*A Characterization of Competition Graphs*by R.D. Dutton and R.C. Brigham*Interval Graphs and Food Webs: A finding and a Problem*by Joel E. Cohen*Using Food Webs In Order To Determine Possible Predictors of Primary and Secondary Extinctions*by Kaitlin A. Ryan (a master's thesis)*The (1,2)-step competition number of a graph*by Kim A.S. Factor, Sarah K. Merz, and Yoshio Sano*On The Hypercompetition Number of Hypergraphs with Maximum Degree at Most Two*by Yoshio Sano*Competition hypergraphs*by Martin Sonntag and Hanns-Martin Teichert*On the hypercompetition numbers of hypergraphs*by Boram Park and Yoshio Sano*Directed Hypergraphs and Applications*by Giorgio Gallo, Giustino Longo, and Stefano Pallottino*On k-hypertournament matrices*by Youngmee Koh and Sangwook Ree- On the Computation of the Competition Number of a Graph by Robert J. Opsut

**Artifacts**

I have included three main artifacts from my REU at Marquette University. These were produced at the end of the summer and the paper is still being edited before we (Dr. Factor and I) submit it for publication. I have placed them here instead of at the end of my lengthy research log, which you will find below.

- Paper
- Please note the paper is still in progress and will be subject to periodic updates until completed.

- Presentation
- Poster

**Project Log**

This is my daily log for what I accomplished. Each day includes an estimate on the number of hours I actually worked on the project or in Cudahy. My activities may include updating the wiki page, attending working lunches, meetings with Dr. Factor, working on various projects with Carissa, or configuring printers; that list is not, by any means, exhaustive. Note: the end-of-week reflections are the last things I do on the last day of each week before I leave the office, naturally.

**Week 1 (5/30/17-6/2/17)**

**Day 1 (5/30/17)**

- Attended MSCS REU orientation
- Met with mentor, Dr. Kim A.S. Factor, and began discussing project details
- Dr. Factor and I discussed goals: having
*BioMath: The Biology and Mathematics of Food Webs*done by Thursday and reading*The (1,2)-step competition graph of a tournament*(by Kim A.S. Factor and Sarah Merz) by next Wednesday - Completed pages 1 through 30 of
*BioMath: The Biology and Mathematics of Food Webs*. I plan on finishing the packet tomorrow and seeing how far I can get through chapter 3 of*Applied Combinatorics*by Fred S. Roberts

**Day 2 (5/31/17)**

- Attended library orientation
- Finished the BioMath packet.
- Began reading
*The (1,2)-step competition graph of a tournament* - Began reading chapter 3 of
*Applied Combinatorics*.

**Day 3 (6/1/17)**

- Attended a talk on good research practices by Dr. Kim Factor
- Filled out direct deposit form
- Met with Dr. Kim Factor and drafted the milestones and discussed moving our lunch with Carissa to Wednesday
- Reviewed milestones and compared them to the program's google calendar
- Printed additional readings for the weekend
- Continued reading chapter 3 of
*Applied Combinatorics* - Continued reading
*The (1,2)-step competition graph of a tournament*

**Day 4 (6/2/17)**

- Uploaded a pdf of my milestones for the summer
- Worked on chapter 3 of
*Applied Combinatorics* - Continued reading
*The (1,2)-step competition graph of a tournament* - Began reviewing other papers
- Created project's Wiki page

**Reflection: End of Week 1**

Concerning my research, this week was relatively uninteresting. That is, I spent most of my time trying to find literature (beyond what Dr. Factor gave me), working through background readings, and finding a good work area. Of those three tasks, I spent most of my time on finding a good work area (it was my top priority); I can happily say I completed this task. I cannot wait to really dig into my research--make it past the background reading.

I accomplished my goals for this week. I attended various meetings, orientations, and finished the BioMath module, among others. I intend on furthering my research this weekend. Specifically, I hope to read multiple papers and complete chapter 3 from *Applied Combinatorics*. Next week, I'll be meeting Carissa on Tuesday, and we'll finally discuss the material Dr. Factor gave us; I'm excited.

Made during Week 2 on Day 3: I was told an hour estimate for each week would be nice. I worked approximately 40 hours this week (I did some stuff on Saturday to make up the missing time from Monday).

**Week 2 (6/5/17-6/9/17)**

**Day 1 (6/5/17)**

- Got in around 10:30AM
- Finished reading chapter 3 from
*Applied Combinatorics* - Created a LaTex document where I am writing my notes on
*Applied Combinatorics*--a link to it is here, please note it is still in progress - Created a LaTex document for teaching Carissa TeX
- Configured the printer in room 410
- Finished formatting my personal wiki and began a rough draft of the project's wiki
- Confirmed a meeting with Carissa tomorrow, 6/6, for 8:30 AM before the ethics training.
- Left around 3:30 PM, estimate for the day is 5 hours

**Day 2 (6/6/17)**

- Attended meeting with Carissa at 8:00AM
- Attended ethical research training at 9:00AM
- Handed off
*Applied Combinatorics*to Carissa for reading on Graph theory - Began reviewing
*The (1,2)-step competition graph*with Carissa at 3:15PM - Made note to include approximate total number of hours in end-of-week reflection
- Left the building at 7:08PM, will be back in two hours...Didn't make it back to Cudahy, finished at the apartment. Estimate for the day is 13.5-14 hours

**Day 3 (6/7/17)**

- Got to Cudahy at 9:00AM, discussed various definitions with Carissa
- Lunch with Dr. Factor around 12
- Received
*The (1,2)-step competition number of a graph*by Factor, Merz, and Sano for reading and began reading it - Began compiling definitions, lemmas, propositions, and theorems with Carissa for an examples pdf
- Made edits to the background section for this project's wiki, will finish them tomorrow
- Left at 6:00pm, estimate for the day is 9 hours

**Day 4 (6/8/17)**

- Got to Cudahy at 10:30
- Finished reading Kaitlin Ryan's master's thesis
- Attended working lunch
- Finished reading The (1,2)-step competition number of a graph by Factor, Merz, and Sano
- Began creating examples for our examples pdf (they will be added tomorrow)
- Drafted potential research questions
- Left lab at 4:40, time estimate is 6 hours

**Day 5 (6/9/17)**

- Arrived at 9:00AM
- Finished making examples pdf with Carissa
- Began formulating definitions for generalization
- Began presentation for the meeting with Dr. Factor on Tuesday
- Began file with BibTeX citations on background readings and paragraphs detailing what they were about
- Finished formatting background section for project's wiki page
- Left around 4:20, approximate time for the day is 7 hours

**Reflection: End of Week 2**

During this week, I worked on reading papers, teaching Carissa LaTeX, and formatting definitions. I met Carissa on Tuesday and our working partnership has been going well. We are good at catching each other's errors and talking through our confusion, if any, regarding various concepts. We made significant progress on finishing the wiki, setting up where our research will go, and are getting ready for our presentation with Dr. Factor on Tuesday. Our research is going well, overall, but could be moving faster.

For my personal research, I have some working ideas. I am turning over a few ideas for various generalizations regarding this project. Dr. Factor would like some working definitions, and I intend on giving her some on Tuesday. Over the weekend, I will work on cementing my some ideas into something more concrete and read other papers concerning hypergraphs. I am very excited at the prospect of proving theorems and actually working on math. If she wants it, I intend on helping Carissa find a general direction to go with her research question, if she doesn't want to join me. That all being said, I only have ideas and no actual thought supporting them; I have what my professor, Sean Raleigh, would call a "highdea."

My hour estimate for this week is 41 hours.

### Weekend 2 (6/10/17 & 6/11/17)

- Began formulating definitions in LaTeX for Dr. Factor

**Week 3 (6/12/17-6/16/17)**

**Day 1 (6/12/17)**

- Arrived around 10:00AM
- Finished presentation with Carissa
- Went through presentation with Carissa
- Edited examples pdf with Carissa
- Finished reading hypergraph, characterization, and other papers
- Left around 4:00 (will come back), approximate hours for today is 6 (so far)
- Worked on TeXing up generalization, total hours is around 9

**Day 2 (6/13/17)**

- Arrived around 10:00am
- Worked on generalizations and example to show Dr. Factor
- Met with Dr. Factor with Carissa at 1:00pm
- Began editing paper
- I now have to defend definitions by Thursday (if possible)
- Left lab at 4:00, hours for the day are 6

**Day 3 (6/14/17)**

- Not feeling well, didn't get in until 11:00am, may go home early--(Note:feeling better)
- Working on generalization edits, needs serious work
- Started and finished ethics training quizzes
- Looked over the remaining modules
- Left office at 6:00pm, approximate time is 7 hours

**Day 4 (6/15/17**

- Began working at 8:30, got in at 11:00
- Met with Dr. Factor at 11:00, was late because I thought it was the working lunch
- Discussed scanning notes over to Dr. Factor, discuss them on Tuesday
- Attended working lunch at 11:30
- Working on edits
- Modified milestones for relevance
- Left around 4:30, approximate hours for the day: 8

**Day 5 (6/16/17)**

- Did not go into the lab today, worked in the apartment
- Read paper at home
- Approximate hours: 6

**Reflection: End of Week 3**

This week, I started interacting with mathematical objects. I worked on my generalization a bit, and I stumbled across some other interesting ideas. I am now looking for definitions concerning hypertournaments in lieu of k-hypertournaments. I'm currently looking at the paper Dr. Factor gave me to look over; it is very dense but at an appropriate level. I have a growing interest in hypergraphs and their colorings. I wonder if I could combine the coloring of hypergraphs and my generalization. I feel as if my "highdea" is becoming more concrete. Lastly, Carissa and I have officially split up, in terms of research questions; she, however, if interested, may join my project. I plan on finishing the edits on that paper before Monday, preferably by the end of tomorrow.

My weekly hour estimate is 36 hours.

**Week 4 (6/19/17-6/23/17)**

**Day 1 (6/19/17)**

- Began at 10:00
- Finished revising paper
- Determined a "hypertournament" is ridiculous
- Question of (i,j)-hypertournaments arose---just a layered (i,j)-step competition graph, probably
- Working various proof ideas
- Working on ideas for colorings and their relationships
- Left building at 2:45, finish researching at 5:30.

Daily hours: 7.5

**Day 2 (6/20/17)**

- Began at 10:00
- Met with Dr. Factor
- Given explicit deadlines for Thursday
- Began reconciling definitions
- Began mini presentation
- Began NP proof
- Finished at 5:30

Daily hours: 7.5

**Day 3 (6/21/17)**

- Began working at 10:00
- Worked on mini presentation
- Worked on definitions
- Worked on NP
- Finished at 8:00

Daily hours: 10

**Day 4 (6/22/17)**

- Began around 9:30
- Worked on mini presentation
- Attended working lunch
- Met with Dr. Factor, obtained approval on my definitions/resolution of definitions
- Worked on TeXing hypergraphs and directed hypergraphs
- Left around 6

Daily hours: 8.5

**Day 5 (6/23/17)**

- Began working at 9:00AM
- Worked on building a hypergraph is LaTeX
- Ended at 11:00AM
- Resumed at 10:00PM and finished at 12:00AM

Daily hours: 5

**Reflection: End of Week 4**

This past week did not feel very productive regarding my research question. I determined my definition is new and interesting enough for pursuit. I also figured out the question regarding whether or not it is NPC is rather simple; hopefully, I'll have a proof by the end of the week. I have also produced enough questions to pursue over the summer. These questions concerning my definition are as follows: is it really just a layering of (*i,j*)-step competition graphs; is its construction NPC; does it have a universal competition number; are there any algebraic connections to this layering at direct products (another 'highdea').

My weekly hour estimate is 38.5 hours.

**Week 5 (6/26/17-6/30/17)**

**Day 1 (6/26/17)**

- Began working at 10:00AM
- Worked on presentation all day (drawing hyperdigraphs in a simply way,
*finally*figured it out) - Ended at 4:00PM
- Resumed at 9:00PM and finished at 12:00AM

Daily hours: 9

**Day 2 (6/27/17)**

- Began at 9:00AM
- Polished presentation and sent to Dr. Factor
- Took a gym break at 11:00AM
- Resumed at 1:30PM
- Made suggested edits to presentation
- Ran into BibTeX problems and worked on those
- Poslished presentation, practiced
- Ended at 7:30

Daily hours: 8 hours

**Day 3 (6/28/17)**

- Began at 11:00AM
- Polished hyperdigraph model
- Practiced presentation with other researchers
- Gave feedback on presentations
- Finished at 10:00PM

Daily hours: 11

**Day 4 (6/29/17)**

- Began at 8:00AM
- Practiced presentations with Carissa
- Listened to mini presentations from 11:00AM to 1:30PM
- Worked on wiki page
- Mapped out project directions (which aspects I'll tackle first)
- Left at 4:00PM

Daily hours: 8

**Day 5 (6/30/17)**

- I took this day off because I was visiting Chicago for the weekend.

Daily hours: 0

**Reflection: End of Week 5**

I spent most of this week preparing my presentation; therefore, my reflection focuses on it. I had a breakthrough on creating simple hyperdigraphs in LaTeX without wild packages, though it is still a pain, which is nice. I created my presentation using a beamer class and scripted it, which I feel was helpful. Concerning my presentation's quality, I'm still waiting for my feedback. In terms of improvements, I feel I could drop the introduction to graphs and digraphs; I think those two slides don't help me. Instead, I could spend more time building the foundation for my definition. I could also improve my conclusion; it feels very rushed when compared to the rest of my presentation. Overall, I think my presentation was fine but has room for improvement.

This past week was not very productive regarding my research. That is, I spent so much time focusing on my presentation that I didn't end up doing actual research, a disappointment. I have so many ideas and questions regarding my definition. Hopefully I get to do some next week.

Weekly hours estimate: 36

**Week 6 (7/3/17-7/7/17)**

**Day 1 (7/3/17)**

- Took this day off for the holidays

Daily hours: 0

**Day 2 (7/4/17)**

- Took this day off for the holidays

Daily hours: 0

**Day 3 (7/5/17)**

- Began at 11:00AM
- Refined definition
- Waited for John Engbers to discuss hyperdigraphs
- Ended at 4:00PM
- Note: this day was short, sadly, but tomorrow will be much longer

Daily hours: 5

**Day 4 (7/6/17)**

- Began at 9:00AM
- Polished definition for Dr. Factor
- Attended working lunch concerning posters
- Had meeting with Dr. Factor and discussed the following
- Definition notation and communication
- Preparing abstract for submission to the YMC
- Briefly discussed poster

- Will send to Dr. Factor tonight
- Finished definition
- Abstract for YMC

- Finished at 7:35PM

Daily hours: 10.5ish

**Day 5 (7/7/17)**

- Began at 9:00AM
- Constructed examples for definition
- Assembled definitions for paper
- Took break at 11:00AM
- Resumed at 12:00PM
- Checked examples for definition
- Sent things in to Dr. Factor
- Finished at 7:00PM

Daily hours: 9

**Reflection: End of Week 6**

I am doing this reflection late because I intended on working over the weekend. Instead, I got distracted studying for the GRE Mathematics Subject Test. Anyways, this week was fairly productive. I completed an abstract for submission to the Young Mathematicians Conference (YMC) at Ohio State University (hopefully I'll be accepted); I prepared an outline for my paper and am working on filling it in; I also began preparing my poster and am working on filling it in. Hopefully, I can get all of my paper and poster done (to the extent that I can right now) and focus solely on math. I am currently working on a uniqueness theorem and characterizing my new definition.

Weekly hours estimate: 24.5

**Week 7 (7/10/17-7/14/17)**

**Day 1 (7/10/17)**

- Began working at 9:00AM
- Began paper outline
- Worked on filling in holes in my paper
- Finished working at 5:00PM

Daily hours: 8

**Day 2 (7/11/17)**

- Began working at 8:00AM
- Filled in notation for my paper
- Finished paper
- Finished at 4:00PM

Daily hours: 8

**Day 3 (7/12/17)**

- Began working at 10:00AM
- Began poster outline
- Filled in poster and made small changes
- Finished 3:30

Daily hours: 5.5

**Day 4 (7/13/17)**

- Began working at 11:00
- Met with Dr. Factor
- Attended working lunch
- Discussed graduate school with Dr. Factor
- Finished poster
- Finished working at 6:00PM

Daily hours: 7

**Day 5 (7/14/17)**

- Began working at 12:00PM
- Tinkered with poster (I changed my color scheme)
- Worked on theorems
- Included potential theorems in my paper
- Proved half of a theorem (one way of an if and only if)

- Ended at 5:00PM

Daily hours: 5

**Reflection: End of Week 7**

I did a lot this week but accomplished very little. That is, I submitted my abstract to the Young Mathematicians Conference, and I polished my poster outline along with my paper outline, too. Completing these three tasks was nice, but they, ultimately, do not mean much if I don't produce enough results. That's why, over the weekend, I completed a proof of a theorem and outlined my proofs for two others. Now, as exciting as this sounds, my definition is, essentially, such a natural extension of Factor and Merz's (i,j)-step competition graph that my proofs follow an almost identical structure to theirs; while this situation makes producing my results incredibly simple, it takes the fun out of it. Hence, I've become a little bored concerning my actual research. That is why I intend on exploring the competition number of hyperdigraphs and showing the NP-completeness of the construction of my object. The former will provide a challenge while the latter is merely interesting.

Weekly hours: approximately 38.5

### Weekend 7 (7/14/17 & 7/16/17)

- Saturday
- I proved the other half of my if and only if (I now "have" one theorem for this summer)
- Experimented with the idea of being "arbitrarily complete"
- I worked about 2 hours

- Sunday
- Resolved how to orient an arbitrarily complete hypergraph
- Looked for definitions concerning arbitrarily complete hypergraphs
- Made notes to ask Dr. Factor about arbitrarily complete hypergraphs
- I worked about 3 hours

**Week 8 (7/17/17-7/21/17)**

**Day 1 (7/17/17)**

- Began at 12:00PM
- Typed theorems
- Figured out theorem 2 proof
- Played with poster and filling in information
- Ended at 5:00PM

Daily hours: 5

**Day 2 (7/18/17)**

- Began at 12:00AM
- Typed proofs to both lemmas
- Wrote proof to theorem
- Got rejected from the YMC :(
- Ended at 5:00PM

Daily hours: 5

**Day 3 (7/19/17)**

- Began at 1:00PM
- Finished rough draft of paper
- Ended at 9:00PM

Daily hours: 8

**Day 4 (7/20/17)**

- Took a break

Daily hours: 0

**Day 5 (7/21/17)**

- Took a break

Daily hours: 0

**Reflection: End of Week 8**

This past week consisted of accomplishing major milestones: I filled in my paper and poster. Establishing the body of my paper required finishing my theorems, and completing my poster followed. I spent most of my week observing my extension of Factor and Merz's paper. That is, I played with what constructed the body of each theorem, and I discovered the reasons behind their paper's structure. I found how they implemented a "story" into their mathematics and told it through natural mathematical progression. I feel I can now delineate between learning mathematics and developing it--though these actions are not mutually exclusive.

Despite making significant progress in my REU milestones, I was rejected from the YMC. I did not expect the YMC committee would accept my proposal for a talk but, if they had, it would have been a nice surprise. I suppose I could try submitting an abstract to another conference but I think I would rather focus on (hopefully) publishing a paper.

Weekly hours: 18

**Week 9 (7/24/17-7/28/17)**

**Day 1 (7/24/17)**

- Began at 12:00PM
- Worked on poster edits
- Ended at 5PM

Daily hours: 5

**Day 2 (7/25/17)**

- Began working at 7:00AM
- Polished poster
- Ended at 12:00PM

Daily hours: 5

**Day 3 (7/26/17)**

- Began at 11:00AM
- Attended working lunch at 11:30AM
- Developed final presentation
- Began editing paper
- Finished at 5:00PM

Daily hours: 6

**Day 4 (7/27/17)**

- Began at 2:00PM
- Finished ethics training (finally)
- Finished at 7:00PM

Daily hours: 5

**Day 5 (7/28/17)**

- Began at 2:00PM
- Polished presentation
- Thought about new theorems
- Ended at 7:00PM

Daily total: 5 hours

**Reflection: End of Week 9**

Nothing interesting really happened this week. This week and next week are either presentations (poster/powerpoint) or preparing for presentations. Hence, I mainly polished the materials I had. Meaning, I made small tweaks to my poster; I worked on some new theorems to add into my paper at a later date; or I practiced my presentation, among other things. I am slightly nervous for the poster presentation because I've never done one before. I've written papers and I've given presentations but I've never done anything related to a poster. However, seeing my poster printed and how well it turned out calmed that anxiety. Lastly, I FINALLY completed those ethics training assignments. Overall, I'm hoping next week goes well!

Weekly hours: 26

**Week 10 (7/31/17-8/4/17)**

**Day 1 (7/31/17)**

- Began at 10:00AM
- Practiced presentation
- Ended at 5:00PM

Daily hours: 7

**Day 2 (8/1/17)**

- Began at 8:45AM
- Attended poster session
- Spoke with Dr. Factor
- Ended at 3:00PM

Daily hours: 6

**Day 3 (8/2/17)**

- Began at 9:00AM
- Practiced presenation
- Attended presentation session
- I presented. You can find my presentation above

- Finished at 2:00PM

Daily hours: 5

**Day 4 (8/3/17)**

- Began at 10:00AM
- Attended presentations
- Ended at 12:00PM

Daily hours: 2

**Day 5 (8/4/17)**

- Began at 10:00AM
- Worked on and off throughout the day/night
- Polishing wiki
- Adding last touches on paper before submitting it as an artifact (will come back to revise)

- Ended at an arbitrary time

Daily hours: 3

**Reflection: End of Week 10**

I don't have much to say. All of my presentations went well. Yes, I wish I could have changed a few things but those changes could only happen if I had more than ten weeks to conduct research. Regarding my time at Marquette, I had a wonderful REU experience. I loved working with my mentor. I discovered a lot about graduate schools, writing mathematics papers, ethical research, and delineating between the development of mathematics and teaching/learning it. I would absolutely do this again.

Weekly hours: 23