## Combinatorial Computing, Assignments CSCI-761, Spring 2021

Upload single pdf with your solutions to each assignment to myCourses, or if infeasible, send it by email to spr@cs.rit.edu with a single pdf attachment containing entire submission. Show the details of your work, give brief reasons for your answers.

### Assignment 1, due Tuesday, February 9

#### Part 1, connecting nauty (25 = 15 + 10 points)

nauty by Brendan McKay
1. Install nauty on your computer, or get other access to nauty package, and learn its basic command-line usage. Try first the functions geng, countg and pickg. In each case '-help' lists available options. Describe briefly your environment.
2. There are exactly 2 nonisomorphic graphs on 10 vertices which have 16 or 17 edges, without cycles of length 4, and with maximum degree 4. Find these graphs using nauty functions.
• Which nauty functions and with what options did you use?
• Print g6-format of the nauty canonical labeling of these two graphs.
• Show 0-1 matrices of these graphs when labeled canonically according to nauty.
zooming in on the two graphs resulting from 2

#### Part 2, no programming (25 = 5 + 10 + 10 points)

Some small graphs.
1. Draw all nonisomorphic graphs on up to 4 vertices.
2. How many distinct (5 x 5) 0-1 matrices are there representing a pentagon? Prove it in just a few lines of common sense reasoning (say by counting the same things in different ways).
3. Draw the two graphs from the second part of Part 1 as nicely as you can.
matrices and graphs for questions 2 and 3 of part 2

### Assignment 2, due Thursday, February 18

(due date extended to 2/20)

#### Part 1, nauty, pipes and graph6 (40 = 10 + 15 + 15 points)

In this part of the assignment, use (nauty, yours or other) functions which read and write graphs in g6-format of graphs. Use pipes in at least some places.
1. In part 2.1 of the previous assignment you found 11 graphs on 4 vertices. Print g6-format of those among them which have no triangles, put them into the file n4.g6.
2. Show that you can read, process and write graphs in g6-format. Write a program which reads graphs from input file I=n[k].g6 and constructs from it the output file O=n[k+1].g6, such that O consists exactly of all canonically labeled graphs, which have (k+1) vertices, have no triangles, and no independent sets of order 5.
• Iterate your program for (k=4;k<14;k++). Which nauty functions and with what options did you use? Make use of some pipes. Include any special script, if any. You may corroborate your results with the contents of table III on page 46 of the paper at position #97 of the list (here are just the 4 needed tables extracted from tabs88.pdf).
• Print canonical g6-format of graphs in n12.g6, n13.g6 and n14.g6.
• Include commented source code you wrote for this assignment (do not include nauty code or any parts of other libraries, but do include any of your scripts using them).
3. [optional] Follow the process of item 2., suitably adjusted, for triangle-free graphs but avoiding K6 instead of K5.

#### Part 2, no programming, just some nauty help (10 points)

For each graph below list generators of its automorphism group and explain why they show up (or not) in your drawing (dreadnaut and countg --a may help). Label your graphs suitably.
1. Draw nicely any graph in n13.g6.
2. Draw nicely the two most symmetric graphs among those in n12.g6.
Sample solution by Hannah Miller.

### Assignment 3, due Tuesday, March 2

(50 = 2*25 points)
1. List all the elements of automorphism groups of two (3,4;8)-graphs (Graph 2 and Graph 3 on the page). Make two listings of permutations for each: as mappings and in cycle notation (thus, 4 listings in total).
2. For classical two-color Ramsey numbers R(s,t), it holds that R(s,t) <= R(s,t-1) + R(s-1,t) for all r, s >= 3. Furthermore, this inequality is strict if both R(s,t-1) and R(s-1,t) are even. Using this, R(s,t) = R(t,s), and R(s,2) = s, derive the best upper bounds you can obtain for R(s,t), for all 3 <= s <= t <= 10. Present the bounds in a table. Mark the entries for which you used the clause "if both R(s,t-1) and R(s-1,t) are even".

Sample solution by Daman Morris,
sample solution by Jiaqing Shen,
sample solution by Xiaoyi Yang.

### Assignment 4, due Tuesday, March 16

1. Draw Cayley graphs of the two automorphism groups of Graph 2 and Graph 3 on this page, using the generators as listed there. Since in these cases all generators are involutions, your Cayley graphs can be shown as undirected graphs (example: Cayley graphs of the group of automorphisms of C5 for three different pairs of generators).
2. Let Jn=Kn-e denote the complete graph Kn with one edge dropped. Trivially, R(J4,J3)=5. It is known that R(J4,J4)=10, R(J4,J5)=13, R(J4,J6)=17 and R(J4,J7)=28. The first open case for the Ramsey numbers of this type is for J4 versus J8, for which the best known bounds are 30 <= R(J4,J8) <= 32 (see pages 12 and 13 of the survey SRN for more details and references).

1. Generate and describe all graphs in (J4,J4;9) and (J4,J5;12).
2. [Optional] Generate and describe all graphs in (J4,J6;16).
3. [Optional] Generate and describe all graphs in (J4,J7;27). Hint: the famous Schläfli graph, which remarkably has 51840 automorphisms, is involved. See the bounds listed in Table IIIa in the survey paper SRN.

Submit a description of what and how you did it. Include source code developed specifically for this assignment.

Sample solution by Daman Morris,
sample solution by Xiaoyi Yang.

### Take-Home Midterm Exam, due Saturday, March 20

Posted on myCourses on Thursday, March 18, at 5pm.
Key on myCourses available from March 23 until March 31.

### Assignment 6, due Thursday, April 8

This assignment is posted as a pdf here, and also the same is on myCourses.

Sample solution by Daman Morris,
sample solution by Zohair Hassan.

### Assignment 7, due Friday, April 30

Let Evrn(G) and Uvrn(G) stand for the existential and universal reconstruction numbers of G, respectively. There are 11 nonisomorphic graphs on 4 vertices, and let us call this set All11. All graphs G referred to below are in All11.

1. List the values of Evrn(G) and Uvrn(G) for all graphs G, aside of their drawings.
2. Give a proof of Evrn(G)=3 for a graph G of your choice.
3. Give all proofs of the values for Evrn(G)>3.
4. Give all proofs of the values for Uvrn(G)=3.
5. Give a proof of Uvrn(G)=4 for a graph G of your choice.

Sample solution by Giovana Puccini.

### Final Exam, online Thu 5/13 - Sat 5/15

To be posted on myCourses Thu 5/13 17:00
Due Sat 5/15 23:59

Back to the course page