1,826 research outputs found
Coloring Drawings of Graphs
We consider face-colorings of drawings of graphs in the plane. Given a
multi-graph together with a drawing in the plane with only
finitely many crossings, we define a face--coloring of to be a
coloring of the maximal connected regions of the drawing, the faces, with
colors such that adjacent faces have different colors. By the -color
theorem, every drawing of a bridgeless graph has a face--coloring. A drawing
of a graph is facially -colorable if and only if the underlying graph is
Eulerian. We show that every graph without degree 1 vertices admits a
-colorable drawing. This leads to the natural question which graphs have
the property that each of its drawings has a -coloring. We say that such a
graph is facially -colorable. We derive several sufficient and necessary
conditions for this property: we show that every -edge-connected graph and
every graph admitting a nowhere-zero -flow is facially -colorable. We
also discuss circumstances under which facial -colorability guarantees the
existence of a nowhere-zero -flow. On the negative side, we present an
infinite family of facially -colorable graphs without a nowhere-zero
-flow. On the positive side, we formulate a conjecture which has a
surprising relation to a famous open problem by Tutte known as the
-flow-conjecture. We prove our conjecture for subcubic and for
-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 the maximal color appears exactly once on the vertices of
. 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 .
Fabrici and G\"oring [Fabrici and Goring 2016] even conjectured that 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 -connected plane graph
admits such a coloring with at most colors.Comment: 5 figure
Vertex coloring of plane graphs with nonrepetitive boundary paths
A sequence is a repetition. A sequence
is nonrepetitive, if no subsequence of consecutive terms of form a
repetition. Let be a vertex colored graph. A path of is nonrepetitive,
if the sequence of colors on its vertices is nonrepetitive. If is a plane
graph, then a facial nonrepetitive vertex coloring of is a vertex coloring
such that any facial path is nonrepetitive. Let denote the minimum
number of colors of a facial nonrepetitive vertex coloring of . Jendro\vl
and Harant posed a conjecture that can be bounded from above by a
constant. We prove that for any plane graph
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
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