389 research outputs found
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
Fine structure of 4-critical triangle-free graphs I. Planar graphs with two triangles and 3-colorability of chains
Aksenov proved that in a planar graph G with at most one triangle, every
precoloring of a 4-cycle can be extended to a 3-coloring of G. We give an exact
characterization of planar graphs with two triangles in that some precoloring
of a 4-cycle does not extend. We apply this characterization to solve the
precoloring extension problem from two 4-cycles in a triangle-free planar graph
in the case that the precolored 4-cycles are separated by many disjoint
4-cycles. The latter result is used in followup papers to give detailed
information about the structure of 4-critical triangle-free graphs embedded in
a fixed surface.Comment: 38 pages, 6 figures; corrections from the review proces
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
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