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 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. In this paper, we study the parameterized complexity of the problem with respect to structural graph parameters. We show that \lcp{} cannot be solved in time $f(w)n^{o(w)}$, unless ETH fails and it can be solved in time $n^{O(w)}$, where $n$ is the size of the input graph, $w$ is the clique-width of the graph and $f$ is an arbitrary function of $w$. Next, we consider the parameters distance to cluster graphs, distance to co-cluster graphs and distance to threshold graphs, which are weaker than the parameter clique-width and show that the problem is fixed-parameter tractable (FPT) with respect to these parameters. Finally, we show that \lcp{} is NP-complete even on bipartite graphs and split graphs.

翻译：根据一个GG$和正整整数美元, 2- Load 颜色问题在于检查是否有2美元彩色 $f:V(G)\rightrow {r,b ⁇ $$$,这样每1美元,至少有1美元边端,两端都有1美元彩色美元。已知问题甚至在普通图表等特殊类别的图表上也是NP-完成的。 Gutn 和 Jones (Inf procent Lett 114:446-449,2014) 表明问题可以通过给最多为7K美元的内核来固定等离差值。 Barbero et al. (Algorithmica 79:21-229,2017) 获得一个小于4K美元头顶端和 $O(k) 边端端端, 改善早期结果。 在本文中, 我们研究与结构图形参数有关的参数的参数比较复杂度复杂性。 我们显示, 距离的参数是, 直径 值 值 和直径 值 值 值 的值 值 无法解 。