An Achievable Low Complexity Encoding Scheme for Coloring Cyclic Graphs

In this paper, we utilize graph coloring for functions captured by cycle graphs to achieve lossless function compression. For functional compression of characteristic graphs or cycles and their OR graphs, we present a feasible scheme for coloring the odd cycles and their OR power graphs. In addition, we calculate the chromatic number for coloring the n-th power of even cycles, which paves the way for calculating the minimum entropy coloring. Moreover, we determine the number of distinct eigenvalues in each OR power and the largest eigenvalue for each power, which allows us to derive a bound on the chromatic number of each OR power of a cycle. Our method provides a proxy for understanding the fundamental limits of compressing cyclic graphs, which has broad applications in various practical use cases of cryptography and computer algebra.