13,659 research outputs found
Choosability in signed planar graphs
This paper studies the choosability of signed planar graphs. We prove that
every signed planar graph is 5-choosable and that there is a signed planar
graph which is not 4-choosable while the unsigned graph is 4-choosable. For
each , every signed planar graph without circuits of length
is 4-choosable. Furthermore, every signed planar graph without circuits of
length 3 and of length 4 is 3-choosable. We construct a signed planar graph
with girth 4 which is not 3-choosable but the unsigned graph is 3-choosable.Comment: We updated the reference lis
Triangulating planar graphs while keeping the pathwidth small
Any simple planar graph can be triangulated, i.e., we can add edges to it,
without adding multi-edges, such that the result is planar and all faces are
triangles. In this paper, we study the problem of triangulating a planar graph
without increasing the pathwidth by much.
We show that if a planar graph has pathwidth , then we can triangulate it
so that the resulting graph has pathwidth (where the factors are 1, 8
and 16 for 3-connected, 2-connected and arbitrary graphs). With similar
techniques, we also show that any outer-planar graph of pathwidth can be
turned into a maximal outer-planar graph of pathwidth at most . The
previously best known result here was .Comment: To appear (without the appendix) at WG 201
Tree-colorable maximal planar graphs
A tree-coloring of a maximal planar graph is a proper vertex -coloring
such that every bichromatic subgraph, induced by this coloring, is a tree. A
maximal planar graph is tree-colorable if has a tree-coloring. In this
article, we prove that a tree-colorable maximal planar graph with
contains at least four odd-vertices.
Moreover, for a tree-colorable maximal planar graph of minimum degree 4 that
contains exactly four odd-vertices, we show that the subgraph induced by its
four odd-vertices is not a claw and contains no triangles.Comment: 18pages,10figure
Nodally 3-connected planar graphs and convex combination mappings
A convex combination mapping of a planar graph is a plane mapping in which
the external vertices are mapped to the corners of a convex polygon and every
internal vertex is a proper weighted average of its neighbours. If a planar
graph is nodally 3-connected or triangulated then every such mapping is an
embedding (Tutte, Floater).
We give a simple characterisation of nodally 3-connected planar graphs, and
generalise the above result to any planar graph which admits any convex
embedding.Comment: 27 pages Latex, 11 postscript figure
A Simple Proof of the Four-Color Theorem
A simpler proof of the four color theorem is presented. The proof was reached
using a series of equivalent theorems. First the maximum number of edges of a
planar graph is obatined as well as the minimum number of edges for a complete
graph. Then it is shown that for the theorem to be false there must exist a
complete planar graph of edges such that . Finally the theorem is
proved to be true by showing that there does not exist a complete planar graph
with .Comment: 4pages, 2 figure
Gr\"obner Basis Convex Polytopes and Planar Graph
Using the Gr\"obner basis of an ideal generated by a family of polynomials we
prove that every planar graph is 4-colorable. Here we also use the fact that
the complete graph of 5 vertices is not included in any planar graph
A Sufficient condition for DP-4-colorability
DP-coloring of a simple graph is a generalization of list coloring, and also
a generalization of signed coloring of signed graphs. It is known that for each
, every planar graph without is 4-choosable.
Furthermore, Jin, Kang, and Steffen \cite{JKS} showed that for each , every signed planar graph without is signed 4-choosable. In
this paper, we show that for each , every planar graph
without is 4-DP-colorable, which is an extension of the above results
On the Circumference of Essentially 4-connected Planar Graphs
A planar graph is essentially -connected if it is 3-connected and every of
its 3-separators is the neighborhood of a single vertex. Jackson and Wormald
proved that every essentially 4-connected planar graph on vertices
contains a cycle of length at least , and this result has
recently been improved multiple times.
In this paper, we prove that every essentially 4-connected planar graph
on vertices contains a cycle of length at least . This
improves the previously best-known lower bound .Comment: 26 page
A Log-space Algorithm for Canonization of Planar Graphs
Graph Isomorphism is the prime example of a computational problem with a wide
difference between the best known lower and upper bounds on its complexity. We
bridge this gap for a natural and important special case, planar graph
isomorphism, by presenting an upper bound that matches the known logspace
hardness [Lindell'92]. In fact, we show the formally stronger result that
planar graph canonization is in logspace. This improves the previously known
upper bound of AC1 [MillerReif'91].
Our algorithm first constructs the biconnected component tree of a connected
planar graph and then refines each biconnected component into a triconnected
component tree. The next step is to logspace reduce the biconnected planar
graph isomorphism and canonization problems to those for 3-connected planar
graphs, which are known to be in logspace by [DattaLimayeNimbhorkar'08]. This
is achieved by using the above decomposition, and by making significant
modifications to Lindell's algorithm for tree canonization, along with changes
in the space complexity analysis.
The reduction from the connected case to the biconnected case requires
further new ideas, including a non-trivial case analysis and a group theoretic
lemma to bound the number of automorphisms of a colored 3-connected planar
graph. This lemma is crucial for the reduction to work in logspace
Partitioning a triangle-free planar graph into a forest and a forest of bounded degree
An -partition of a graph is a vertex-partition into
two sets and such that the graph induced by is a forest and the
one induced by is a forest with maximum degree at most . We prove that
every triangle-free planar graph admits an -partition.
Moreover we show that if for some integer there exists a triangle-free
planar graph that does not admit an -partition, then it
is an NP-complete problem to decide whether a triangle-free planar graph admits
such a partition.Comment: 16 pages, 12 figure
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