Star coloring

The star chromatic number of the Dyck graph is 4, although its chromatic number is 2.

In the mathematical field of graph theory, a star coloring of a graph Template:Mvar 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.Template:Sfnp Star coloring has been introduced by Template:Harvtxt.Template:Sfnp The star chromatic number Template:Tmath of Template:Mvar is the fewest colors needed to star color Template:Mvar.

In special classes of graphs

Template:Harvtxt observed that the star chromatic number is bounded for planar graphs.Template:Sfnp More precisely, the star chromatic number of planar graphs is at most 20, and some planar graphs have star chromatic number at least 10.Template:Sfnp More generally, the star chromatic number is bounded on every proper minor closed class.Template:Sfnp This result has been generalized to all low-tree-depth colorings (standard coloring and star coloring being low-tree-depth colorings with respective parameter 1 and 2).Template:Sfnp

For every graph of maximum degree Template:Tmath the star chromatic number is Template:Tmath There exist graphs for which this bound is close to tight: they have star chromatic number Template:TmathTemplate:Sfnp

Complexity

It is NP-complete to determine whether ${\chi }_s(G)\le 3$, even when G is a graph that is both planar and bipartite.Template:Sfnp Finding an optimal star coloring is NP-hard even when G is a bipartite graph.Template:Sfnp

Related concepts

Star coloring is the special case for $q=3$ of $q$-centered coloring, colorings in which every connected subgraph either uses at least $q$ colors or has at least one color that is used for exactly one vertex. For such a coloring, a connected subgraph with only two colors must be a star, with the vertex of a unique color at its center. There can be no edges between the remaining vertices in the component, because they would form two-vertex connected subgraphs without a uniquely used color.

Another 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 Template:Mvar by Template:Tmath, we have that Template:Tmath, and in fact every star coloring of Template:Mvar is an acyclic coloring. In the other direction, Template:Tmath so each of the two kinds of chromatic number is bounded if and only if the other one is.Template:Sfnp

References

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External links

• Star colorings and acyclic colorings (1973), present at the Research Experiences for Graduate Students (REGS) at the University of Illinois, 2008.