On Structural Parameterizations of Load Coloring

Given a graph G and a positive integer k, the 2-Load Coloring problem is to check whether there is a 2-coloring \(f:V(G) \rightarrow \{r,b\}\) of G such that for every \(i \in \{r,b\}\), there are at least k edges with both end vertices colored i. It is known that the problem is \(\mathsf {NP} \)-complete even on special classes of graphs like regular graphs. Gutin and Jones (Inf Process Lett 114:446-449, 2014) showed that the problem is fixed-parameter tractable by giving a kernel with at most 7k vertices. Barbero et al. (Algorithmica 79:211-229, 2017) obtained a kernel with less than 4k vertices and O(k) edges, improving the earlier result.