Unique-maximum coloring of plane graphs

Abstract

A unique-maximum k-coloring with respect to faces of a plane graph G is a coloring with colors 1,. .. , k so that, for each face α of G, the maximum color occurs exactly once on the vertices of α. We prove that any plane graph is unique-maximum 3-colorable and has a proper unique-maximum coloring with 6 colors.

On facial unique-maximum (edge-)coloring

“…First, we recall a theorem, which is the main tool used in [5], and will prove helpful also in proving our results.…”

“…A proper coloring of a graph embedded on some surface, where colors are integers and every face has a unique vertex colored with a maximal color, is called a facial uniquemaximum coloring or FUM-coloring for short (Wendland uses the notion capital coloring instead). This type of coloring was first studied by Fabrici and Göring [5]. The main motivation for their research comes from the unique-maximum coloring (also known as ordered coloring), defined as a coloring where there is only one vertex colored with the maximal color on every path in a graph.…”

Facial unique-maximum colorings of plane graphs with restriction on big vertices

“…This conjecture was disproven in the general case by the authors [5]. Fabrici and Göring [4] proved that for any plane graph G, χ fum (G) ≤ 6, while Wendland [8] improved the upper bound to 5. Andova, Lidický, Lužar, and Škrekovski [1] proved that if G is a subcubic or outerplane graph, χ fum (G) ≤ 4.…”

A counterexample to a conjecture on facial unique-maximal colorings

“…The cornerstone of graph colorings is the Four Color Theorem stating that every planar graph can be properly colored using at most four colors [2]. Fabrici and Göring [4] proposed the following strengthening of the Four Color Theorem.…”

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A facial unique-maximum coloring of a plane graph is a proper vertex coloring by natural numbers where on each face α the maximal color appears exactly once on the vertices of α. Fabrici and Göring [4] proved that six colors are enough for any plane graph and conjectured that four colors suffice. This conjecture is a strengthening of the Four Color theorem.

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