Contains some initial work on edge colorings for detour length paths. Thus far, we successfully check for k-color connectivity, determine minimum detour numbers, and view chord additions as permutations of distance between chord incident vertices. Thus far, and algorithmic approach to minimally k-color connecting a graph has not been implemented, and the proposed solution cannot be determined to halt unless the k-color connection number of any graph is less than 2k-1.
Contains some initial work on edge colorings for detour length paths. Thus far, we successfully check for -color connectivity, determine minimum detour numbers, and view chord additions as permutations of distance between chord incident vertices. Thus far, and algorithmic approach to minimally -color connecting a graph has not been implemented, and the proposed solution cannot be determined to halt unless the -color connection number of any graph is less than .
A number of intractable problems arise as we begin exploring this topic. Rainbow-connection numbers (Li, Sun 2012) have been determined to be NP-Hard, likewise the computation of detour matrices has no efficient algorithm; the problem of -color connecting a graph relies on both rainbow-connections, and minimum detour numbers. Regarding the programs, the wolfram package makes use of a graph isomorphism module which separates chorded cycles into equivalence classes; this is done using a function map, making it ideal for usage in parallel. The python files generate a set of all chorded cycles and separate them into classes by there detour number.
An approximation of the -color connection number can be generated using DetourModule.wl, although that package has bugs. It uses all paths of length or greater to color the graph, depending on how frequently a particular edge occurs in a length path.
Interesting Visualizations
While technically out of the scope of the project, I explored some different avenues that led to some interesting images. It became clear that while two chords were responsible for increased detour number of a graph, only certain combinations of these chords would work. We define a new graph, where edges become vertices, and these vertices are adjacent if their presence in the original graph results in a path of length . While the graphs themselves didn't illustrate much, we can see patterns emerge from the adjacency matrices.
where
where
where
About
Contains some initial work on edge colorings for detour length paths. Thus far, we successfully check for k-color connectivity, determine minimum detour numbers, and view chord additions as permutations of distance between chord incident vertices. Thus far, and algorithmic approach to minimally k-color connecting a graph has not been implemented, and the proposed solution cannot be determined to halt unless the k-color connection number of any graph is less than 2k-1.
