Star coloring



In the mathematical field of graph theory, a star coloring of a graph $G$ is a (proper) vertex coloring in which every path on four vertices uses at least three distinct colors. Equivalently, in a star coloring, the induced subgraphs formed by the vertices of any two colors has connected components that are star graphs. Star coloring has been introduced by. The star chromatic number $\chi_s(G)$ of $G$ is the fewest colors needed to star color $G$.

One generalization of star coloring is the closely related concept of acyclic coloring, where it is required that every cycle uses at least three colors, so the two-color induced subgraphs are forests. If we denote the acyclic chromatic number of a graph $G$ by $\chi_a(G)$, we have that $\chi_a(G) \leq \chi_s(G)$, and in fact every star coloring of $G$ is an acyclic coloring.

The star chromatic number has been proved to be bounded on every proper minor closed class by. This results was further generalized by to all low-tree-depth colorings (standard coloring and star coloring being low-tree-depth colorings with respective parameter 1 and 2).

Complexity
It was demonstrated by that it is NP-complete to determine whether $$\chi_s(G) \leq 3$$, even when G is a graph that is both planar and bipartite. showed that finding an optimal star coloring is NP-hard even when G is a bipartite graph.