Vertex-partitions of 2-edge-colored graphs

Abstract: A {\bf $\mathbf{k}$-majority coloring} of a digraph $D=(V,A)$ is a coloring of $V$ with $k$ colors so that each vertex $v\in V$ has at least as many out-neighbours of color different from its own color as it has out-neighbours with the same color as itself. Majority colorings have received much attention in the last years and many interesting open problems remain. Inspired by this and the fact that digraphs can be modelled via 2-edge-colored graphs we study several problems concerning vertex partitions of 2-edge-colored graphs. In particular we study vertex partitions with the property that for each $c=1,2$ every vertex $v$ has least as many edges of colour $c$ to vertices outside the set it belongs to as it has to vertices inside its own set. We call such a vertex partition with $k$ sets a {\bf $\mathbf{k}$-majority partition. Among other things we show that every 2-edge-coloured graph has a 4-majority partition and that it is NP-complete to decide whether a 2-edge-coloured graph has a 3-majority partition. We also apply probabilistic tools to show that every $2$-edge-colored graph $G$ of minimum color-degree $\delta$ and maximum degree $\Delta \le \frac{e<sup>{\delta/18}}{9\delta}-2$ has a balanced majority $3$-partition.