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

A facial unique-maximum coloring of a plane graph is a proper coloring of the vertices using positive integers such that each face has a unique vertex that receives the maximum color in that face. Fabrici and G\"{o}ring (2016) proposed a strengthening of the Four Color Theorem conjecturing that all plane graphs have a facial unique-maximum coloring using four colors. This conjecture has been disproven for general plane graphs and it was shown that five colors suffice. In this paper we show that plane graphs, where vertices of degree at least four induce a star forest, are facially unique-maximum 4-colorable. This improves a previous result for subcubic plane graphs by Andova, Lidick\'y, Lu\v{z}ar, and \v{S}krekovski (2018). We conclude the paper by proposing some problems.Comment: 8 pages, 5 figure

A counterexample to a conjecture on facial unique-maximal colorings

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. Wendland [6] later decreased the upper bound from six to five. In this note, we disprove the conjecture by giving an infinite family of counterexamples. s we conclude that facial unique-maximum chromatic number of the sphere is five

On facial unique-maximum (edge-)coloring

A facial unique-maximum coloring of a plane graph is a vertex coloring where on each face $\alpha$ the maximal color appears exactly once on the vertices of $\alpha$. 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 G\"oring [Fabrici and Goring 2016] 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.Comment: 5 figure

The Four Color Problem: The Journey to a Proof and the Results of the Study

The four color problem was one of the most difficult to prove problems for 150 years. It took several failed proofs and advancement in technology and techniques for the final proof to become possible. Some notable men include De Morgan first writing about the problem, Kempe giving the first proof, Heawood showing the flaws in Kempe’s work as well as making advancements of his own. The first actual proof of the problem is then discussed, as well as it’s shortcomings and the work done by other mathematicians to show improvements on them. The total of this work has lead to numerous great leaps in mathematics including the creation of the branch known as graph theory. This one problem also revolutionized proof writing, being the first to use a computer as an essential part of the proving process

Coloring problems in graph theory

In this thesis, we focus on variants of the coloring problem on graphs. A coloring of a graph $G$ is an assignment of colors to the vertices. A coloring is proper if no two adjacent vertices are assigned the same color. Colorings are a central part of graph theory and over time many variants of proper colorings have been introduced. The variants we study are packing colorings, improper colorings, and facial unique-maximum colorings. A packing coloring of a graph $G$ is an assignment of colors $1, \ldots, k$ to the vertices of $G$ such that the distance between any two vertices that receive color $i$ is greater than $i$. A $(d_1, \ldots, d_k)$-coloring of $G$ is an assignment of colors $1, \ldots, k$ to the vertices of $G$ such that the distance between any two vertices that receive color $i$ is greater than $d_i$. We study packing colorings of multi-layer hexagonal lattices, improving a result of Fiala, Klav\v{z}ar, and Lidick\\u27{y}, and find the packing chromatic number of the truncated square lattice. We also prove that subcubic planar graphs are $(1, 1, 2, 2, 2)$-colorable. A facial unique-maximum coloring of $G$ is an assignment of colors $1, \ldots, k$ to the vertices of $G$ such that no two adjacent vertices receive the same color and the maximum color on a face appears only once on that face. We disprove a conjecture of Fabrici and G\ {o}ring that plane graphs are facial unique-maximum $4$-colorable. Inspired by this result, we also provide sufficient conditions for the facial unique-maximum $4$-colorability of a plane graph. A $\{ 0, p \}$-coloring of $G$ is an assignment of colors $0$ and $p$ to the vertices of $G$ such that the vertices that receive color $0$ form an independent set and the vertices that receive color $p$ form a linear forest. We will explore $\{ 0, p \}$-colorings, an offshoot of improper colorings, and prove that subcubic planar $K_4$-free graphs are $\{ 0, p \}$-colorable. This result is a corollary of a theorem by Borodin, Kostochka, and Toft, a fact that we failed to realize before the completion of our proof