75 research outputs found
Partial duality and closed 2-cell embeddings
In 2009 Chmutov introduced the idea of partial duality for embeddings of
graphs in surfaces. We discuss some alternative descriptions of partial
duality, which demonstrate the symmetry between vertices and faces. One is in
terms of band decompositions, and the other is in terms of the gem
(graph-encoded map) representation of an embedding. We then use these to
investigate when a partial dual is a closed 2-cell embedding, in which every
face is bounded by a cycle in the graph. We obtain a necessary and sufficient
condition for a partial dual to be closed 2-cell, and also a sufficient
condition for no partial dual to be closed 2-cell.Comment: 16 pages, 6 figures, 1 table. To appear in "Journal of combinatorics
Orientable embeddings and orientable cycle double covers of projective-planar graphs
In a closed 2-cell embedding of a graph each face is homeomorphic to an open
disk and is bounded by a cycle in the graph. The Orientable Strong Embedding
Conjecture says that every 2-connected graph has a closed 2-cell embedding in
some orientable surface. This implies both the Cycle Double Cover Conjecture
and the Strong Embedding Conjecture. In this paper we prove that every
2-connected projective-planar cubic graph has a closed 2-cell embedding in some
orientable surface. The three main ingredients of the proof are (1) a surgical
method to convert nonorientable embeddings into orientable embeddings; (2) a
reduction for 4-cycles for orientable closed 2-cell embeddings, or orientable
cycle double covers, of cubic graphs; and (3) a structural result for
projective-planar embeddings of cubic graphs. We deduce that every
2-edge-connected projective-planar graph (not necessarily cubic) has an
orientable cycle double cover.Comment: 16 pages, 3 figure
The Chv\'atal-Erd\H{o}s condition for prism-Hamiltonicity
The prism over a graph is the cartesian product . It is known
that the property of having a Hamiltonian prism (prism-Hamiltonicity) is
stronger than that of having a -walk (spanning closed walk using every
vertex at most twice) and weaker than that of having a Hamilton path. For a
graph , it is known that , where is
the independence number and is the connectivity, imples existence
of a -walk in , and the bound is sharp. West asked for a bound on in terms of guaranteeing prism-Hamiltonicity. In this paper
we answer this question and prove that implies the
stronger condition, prism-Hamiltonicity of .Comment: 7 pages, no figure
Criticality of counterexamples to toroidal edge-hamiltonicity
A well-known conjecture of Gr\"unbaum and Nash-Williams proposes that
4-connected toroidal graphs are hamiltonian. The corresponding results for
4-connected planar and projective-planar graphs were proved by Tutte and by
Thomas and Yu, respectively, using induction arguments that proved a stronger
result, that every edge is on a hamilton cycle. However, this stronger property
does not hold for 4-connected toroidal graphs: Thomassen constructed
counterexamples. Thus, the standard inductive approach will not work for the
torus. One possible way to modify it is by characterizing the situations where
some edge is not on a hamilton cycle. We provide a contribution in this
direction, by showing that the obvious generalizations of Thomassen's
counterexamples are critical in a certain sense.Comment: 8 page
One-way infinite 2-walks in planar graphs
We prove that every 3-connected 2-indivisible infinite planar graph has a
1-way infinite 2-walk. (A graph is 2-indivisible if deleting finitely many
vertices leaves at most one infinite component, and a 2-walk is a spanning walk
using every vertex at most twice.) This improves a result of Timar, which
assumed local finiteness. Our proofs use Tutte subgraphs, and allow us to also
provide other results when the graph is bipartite or an infinite analog of a
triangulation: then the prism over the graph has a spanning 1-way infinite
path.Comment: 23 pages, 4 figure
Edge-outer graph embedding and the complexity of the DNA reporter strand problem
In 2009, Jonoska, Seeman and Wu showed that every graph admits a route for a
DNA reporter strand, that is, a closed walk covering every edge either once or
twice, in opposite directions if twice, and passing through each vertex in a
particular way. This corresponds to showing that every graph has an
\emph{edge-outer embedding}, that is, an orientable embedding with some face
that is incident with every edge. In the motivating application, the objective
is such a closed walk of minimum length. Here we give a short algorithmic proof
of the original existence result, and also prove that finding a shortest length
solution is NP-hard, even for -connected cubic (-regular) planar graphs.
Independent of the motivating application, this problem opens a new direction
in the study of graph embeddings, and we suggest new problems emerging from it.Comment: 16 pages, 7 figures, minor revisio
Toughness and prism-hamiltonicity of -free graphs
The \emph{prism} over a graph is the product , i.e., the
graph obtained by taking two copies of and adding a perfect matching
joining the two copies of each vertex by an edge. The graph is called
\emph{prism-hamiltonian} if it has a hamiltonian prism. Jung showed that every
-tough -free graph with at least three vertices is hamiltonian. In this
paper, we extend this to observe that for a -free graph has a
spanning \emph{-walk} (closed walk using each vertex at most times) if
and only if it is -tough. As our main result, we show that for the
class of -free graphs, the three properties of being prism-hamiltonian,
having a spanning -walk, and being -tough are all equivalent
Toughness and spanning trees in -minor-free graphs
For an integer , a -tree is a tree with maximum degree at most .
More generally, if is an integer-valued function on vertices, an -tree
is a tree in which each vertex has degree at most . Let denote
the number of components of a graph . We show that if is a connected
-minor-free graph and
then has a spanning -tree. Consequently, if is a
-tough -minor-free graph, then has a spanning -tree.
These results are stronger than results for general graphs due to Win (for
-trees) and Ellingham, Nam and Voss (for -trees). The -minor-free
graphs form a subclass of planar graphs, and are identical to graphs of
treewidth at most , and also to graphs whose blocks are series-parallel. We
provide examples to show that the inequality above cannot be relaxed by adding
to the right-hand side, and also to show that our result does not hold for
general planar graphs. Our proof uses a technique where we incorporate
toughness-related information into weights associated with vertices and
cutsets.Comment: Proposition 2.3 in v1 was incorrect; this has been fixed. 25 pages, 1
figur
Minimal quadrangulations of surfaces
A quadrangular embedding of a graph in a surface , also known as a
quadrangulation of , is a cellular embedding in which every face is
bounded by a -cycle. A quadrangulation of is minimal if there is no
quadrangular embedding of a (simple) graph of smaller order in . In
this paper we determine , the order of a minimal quadrangulation of
a surface , for all surfaces, both orientable and nonorientable.
Letting denote the sphere and the Klein bottle, we prove that
, and for all other surfaces , where
is the Euler characteristic. Our proofs use a `diagonal
technique', introduced by Hartsfield in 1994. We explain the general features
of this method.Comment: 25 pages, 20 figure
Quadrangular embeddings of complete graphs and the Even Map Color Theorem
Hartsfield and Ringel constructed orientable quadrangular embeddings of the
complete graph for , and nonorientable ones for and . These provide minimal quadrangulations of their
underlying surfaces. We extend these results to determine, for every complete
graph , , the minimum genus, both orientable and nonorientable,
for the surface in which has an embedding with all faces of degree at
least , and also for the surface in which has an embedding with all
faces of even degree. These last embeddings provide sharpness examples for a
result of Hutchinson bounding the chromatic number of graphs embedded with all
faces of even degree, completing the proof of the Even Map Color Theorem. We
also show that if a connected simple graph has a perfect matching and a
cycle then the lexicographic product has orientable and nonorientable
quadrangular embeddings; this provides new examples of minimal
quadrangulations.Comment: Version 2 is a greatly expanded version of the paper with four new
authors (Lawrencenko, Chen, Hartsfield, Yang). It includes results on
quadrangular or nearly quadrangular embeddings for all complete graphs. It
also includes an application to a coloring result. This version contains some
details omitted from the version that will be submitted for publication. 26
pages; 6 figure
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