Motions on n-Simplex Graphs with m-value memory

TL;DR

This paper explores moves on n-simplex graphs and develops algorithms to determine graph colorability, including 3-colorability, by analyzing label transformations and solution counts.

Contribution

It introduces a new framework for analyzing moves on n-simplex graphs and provides algorithms for graph coloring problems.

Findings

Solved the move transformation problem for certain graph classes

Developed an algorithm to determine (n+1)-colorability

Established methods to count distinct solutions

Abstract

We introduce the idea of an n-simplex graph and games upon simplicial complexes. We then define moves on a labeled graph and pose the problem of whether given two labelings of a graph it is possible to change one into another via these moves. We then solve the problem for a given class of graphs. Once having found a solution for a given class of graphs we determine the number of different solutions that exist. We then use this to find an algorithm to determine whether a graph is (n+1)-colorable, and in particular, whether it is 3-colorable.