2,990 research outputs found
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
Minimal counterexamples and discharging method
Recently, the author found that there is a common mistake in some papers by
using minimal counterexample and discharging method. We first discuss how the
mistake is generated, and give a method to fix the mistake. As an illustration,
we consider total coloring of planar or toroidal graphs, and show that: if
is a planar or toroidal graph with maximum degree at most , where
, then the total chromatic number is at most .Comment: 8 pages. Preliminary version, comments are welcom
Defective 3-Paintability of Planar Graphs
A -defective -painting game on a graph is played by two players:
Lister and Painter. Initially, each vertex is uncolored and has tokens. In
each round, Lister marks a chosen set of uncolored vertices and removes one
token from each marked vertex. In response, Painter colors vertices in a subset
of which induce a subgraph of maximum degree at most . Lister
wins the game if at the end of some round there is an uncolored vertex that has
no more tokens left. Otherwise, all vertices eventually get colored and Painter
wins the game. We say that is -defective -paintable if Painter has a
winning strategy in this game. In this paper we show that every planar graph is
3-defective 3-paintable and give a construction of a planar graph that is not
2-defective 3-paintable.Comment: 21 pages, 11 figure
- …