On facial unique-maximum (edge-)coloring

Abstract

A facial unique-maximum coloring of a plane graph is a vertex coloring where on each face α the maximal color appears exactly once on the vertices of α. If the coloring is required to be proper, then the upper bound for the minimal number of colors required for such a coloring is set to 5. Fabrici and Göring [5] even conjectured that 4 colors always suffice. Confirming the conjecture would hence give a considerable strengthening of the Four Color Theorem. In this paper, we prove that the conjecture holds for subcubic plane graphs, outerplane graphs and plane quadrangulations. Additionally, we consider the facial edge-coloring analogue of the aforementioned coloring and prove that every 2-connected plane graph admits such a coloring with at most 4 colors.

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

“…In this paper we show that plane graphs, where vertices of degree at least four induce a star forest, are facially uniquemaximum 4-colorable. This improves a previous result for subcubic plane graphs by Andova, Lidický, Lužar, and Škrekovski (2018). We conclude the paper by proposing some problems.…”

“…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

“…Moreover, removing the edge a 4 a 2 from the graph in Figure 1 gives a disconnected graph with maximum degree 4 that does not have a FUM-coloring with colors in {1, 2, 3, 4}. Recall that Andova et al [1] showed that maximum degree 3 suffices.…”

“…Wendland [6] decreased the upper bound to 5 for all plane graphs. Andova, Lidický, Lužar, and Škrekovski [1] showed that 4 colors suffice for outerplanar graphs and for subcubic plane graphs. Wendland [6] also considered the list coloring version of the problem, where he was able to prove the upper bound 7 and conjectured that lists of size 5 are sufficient.…”