Balanced coloring
A signed graph, denoted (G, 𝜎 ), is a graph G together with an assignment 𝜎 which assigns to each edge either a positive or a negative sign. Sign of a cycle or a closed walk is the product of the signs of its edges. A signed graph is balanced if it induces no negative cycle.
Balanced coloring of signed graph with no negative loop is to partition (or cover) vertices of G such that each part induces a balanced set, in other word no part induces a negative cycle on its own.
A general question then is to bound the balanced chromatic number of signed graphs satisfying structural properties. For example, the four-color theorem can be restated as: every planar signed graph with no negative loop (allowing parallel edges) admits a balanced 4-coloring. It follows from that Euler formula, or more precisely, from the fact of being 5-degenerate that planar signed simple graphs admit balanced 3-coloring. A first example of planar signed simple graph needing 3 colors in any balanced coloring was built by Kardos and Narboni.
Various structural properties lead to different developments: one may forbid a minor (both in the sense of graphs and signed graphs), may forbid topological minor, may forbid induced subgraph, etc.
A fractional approach to the question leads to the study of signed Kneser graphs, signed Borsuk graph etc.
General this a rich project with various directions for development and can be adapted with the interest an interested student.
Some reference to read:
Complex and homomorphic chromatic number of signed planar simple graphs