Combinatorial Computing, Assignments
CSCI-761, Spring 2026

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Assignment 1, due Friday, January 30

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

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.

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

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.

Assignment 2, due Monday, February 9

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

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 #109 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. above, suitably adjusted, for triangle-free graphs but avoiding K₆ instead of K₅ in the complement.

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

1. Draw nicely any graph in n13.g6.
2. Draw nicely the two most symmetric graphs among those in n12.g6.

Assignment 3, due Tuesday, February 24

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).

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".

   List the upper bounds on R(s,s) implied by R(s,s) <= choose(2s-2,s-1) for 3 <= s <= 10. List the upper bounds on R(s,s) for 3 <= s <= 10 implied by the result of Campos-Griffith-Morris-Sahasrabudhe, as on slide 6 of Boca Raton 2025, with epsilon = 0.2.

3. Consider the lower bound on R(s,s) in Theorem 1 of lecture 6 by Jacob Fox. List the lower bounds on R(s,s) for 3 <= s <=10 as in the theorem, and for each s the best which can be obtained numerically from 2*choose(n,s)/2^choose(s,2) < 1.

Assignment 4, due Thursday, March 5

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).

Assignment 5, due Tuesday, March 31

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Final Exam, Thu 4/30

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