Coloring Drawings of Graphs

We consider face-colorings of drawings of graphs in the plane. Given a multi-graph $G$ together with a drawing $\Gamma(G)$ in the plane with only finitely many crossings, we define a face-$k$-coloring of $\Gamma(G)$ to be a coloring of the maximal connected regions of the drawing, the faces, with $k$ colors such that adjacent faces have different colors. By the $4$-color theorem, every drawing of a bridgeless graph has a face-$4$-coloring. A drawing of a graph is facially $2$-colorable if and only if the underlying graph is Eulerian. We show that every graph without degree 1 vertices admits a $3$-colorable drawing. This leads to the natural question which graphs $G$ have the property that each of its drawings has a $3$-coloring. We say that such a graph $G$ is facially $3$-colorable. We derive several sufficient and necessary conditions for this property: we show that every $4$-edge-connected graph and every graph admitting a nowhere-zero $3$-flow is facially $3$-colorable. We also discuss circumstances under which facial $3$-colorability guarantees the existence of a nowhere-zero $3$-flow. On the negative side, we present an infinite family of facially $3$-colorable graphs without a nowhere-zero $3$-flow. On the positive side, we formulate a conjecture which has a surprising relation to a famous open problem by Tutte known as the $3$-flow-conjecture. We prove our conjecture for subcubic and for $K_{3,3}$-minor-free graphs.Comment: 24 pages, 17 figure

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

Vertex coloring of plane graphs with nonrepetitive boundary paths

A sequence $s_1,s_2,...,s_k,s_1,s_2,...,s_k$ is a repetition. A sequence $S$ is nonrepetitive, if no subsequence of consecutive terms of $S$ form a repetition. Let $G$ be a vertex colored graph. A path of $G$ is nonrepetitive, if the sequence of colors on its vertices is nonrepetitive. If $G$ is a plane graph, then a facial nonrepetitive vertex coloring of $G$ is a vertex coloring such that any facial path is nonrepetitive. Let $\pi_f(G)$ denote the minimum number of colors of a facial nonrepetitive vertex coloring of $G$. Jendro\vl and Harant posed a conjecture that $\pi_f(G)$ can be bounded from above by a constant. We prove that $\pi_f(G)\le 24$ for any plane graph $G$

Third case of the Cyclic Coloring Conjecture

The Cyclic Coloring Conjecture asserts that the vertices of every plane graph with maximum face size D can be colored using at most 3D/2 colors in such a way that no face is incident with two vertices of the same color. The Cyclic Coloring Conjecture has been proven only for two values of D: the case D=3 is equivalent to the Four Color Theorem and the case D=4 is equivalent to Borodin's Six Color Theorem, which says that every graph that can be drawn in the plane with each edge crossed by at most one other edge is 6-colorable. We prove the case D=6 of the conjecture

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