Software

There are three main software components.

• SCI_magma.txt is a list of Magma commands to discover the coefficient of a polynomial to use in a Combinatorial Nullstellensatz argument. (Written by Phil DeOrsey)
• SCI_Sage_code.sage is a list of Sage commands to discover the coefficient of a polynomial to use in a Combinatorial Nullstellensatz argument. (Written by Stephen G. Hartke)
• StrongEdgeColorings.sws is a Sage worksheet that contains many helpful routines, including constructing the configurations. It also tests the base cases of the main theorem regarding strong 5-edge-colorings.
• strong_edge_reducible.c is a C program that tests if a configuration is reducible.

  To compile, place this file in the nauty/ folder after unpacking the nauty source code, available at http://cs.anu.edu.au/~bdm/nauty/

  Compile using the following command:

  gcc -o strong_edge_reducible strong_edge_reducible.c nauty.o nausparse.o nautil.o gtools.o

  Then, run : strong_edge_reducible.exe < configurations.txt

  or (for 7 colors) : strong_edge_reducible.exe -k 7 < configurations.txt

• YColoring.sws is a Sage worksheet that determines that no $3$-vertex $v$ can have $N_3(v)$ consist entirely of vertices $u$ with $|\operatorname{Resp}(u)| \geq 12$. (Written by Bernard Lidický)

Data

• configurations.txt is the input file for strong_edge_reducible.exe containing all configurations, reducible or not.
• ser_output.txt is the output file reporting which configurations are reducible or not.
• caterpillars-d4.txt is the input file for strong_edge_reducible.exe containing a list of $(t,4)$-caterpillars.
• caterpillars-d5.txt is the input file for strong_edge_reducible.exe containing a list of $(t,5)$-caterpillars.
• caterpillars-d6.txt is the input file for strong_edge_reducible.exe containing a list of $(t,6)$-caterpillars.
• reducible-d4y.txt is the input file for strong_edge_reducible.exe containing a list of $Y_4(t_1,t_2,t_3)$ configurations.
• reducible-d4y-out.txt is the output file for strong_edge_reducible.exe -k 7 < reducible-d4y.txt demonstrating which $Y_4(t_1,t_2,t_3)$ configurations are 7-reducible.