Thesis

Star coloring planar graphs

The focus of this thesis is star coloring planar graphs. A star coloring of a planar graph is a proper coloring in which no path on four vertices is colored with just two colors. A graph G is said to be k-star-colorable if G has a star coloring with at most k colors. The fewest number of colors needed to star color a graph G is called the star chromatic number of G. It is known that all planar graphs of girth at least seven can be star colored using at most 9 colors. We prove that all planar graphs of girth at least seven can be star colored using at most 7 colors. Also, we improve upon the current known bounds for star colorings of families of planar graphs of girth at least eight. It is known that there exists a planar graph that requires at least 10 colors to star color. We prove that there are planar bipartite graphs requiring at least 8 colors to star color, and that there are planar graphs of girth 5 requiring at least 6 colors to star color. Finally, we prove that there are planar graphs of girth 6 requiring at least 5 colors to star color. Keywords: graphs, star-coloring, planar, girth, outerplanar

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