12 research outputs found
Irreducible Triangulations are Small
A triangulation of a surface is \emph{irreducible} if there is no edge whose
contraction produces another triangulation of the surface. We prove that every
irreducible triangulation of a surface with Euler genus has at most
vertices. The best previous bound was .Comment: v2: Referees' comments incorporate
Line graphs and -geodesic transitivity
For a graph , a positive integer and a subgroup G\leq
\Aut(\Gamma), we prove that is transitive on the set of -arcs of
if and only if has girth at least and is
transitive on the set of -geodesics of its line graph. As applications,
we first prove that the only non-complete locally cyclic -geodesic
transitive graphs are the complete multipartite graph and the
icosahedron. Secondly we classify 2-geodesic transitive graphs of valency 4 and
girth 3, and determine which of them are geodesic transitive
Difference of Facial Achromatic Numbers between Two Triangular Embeddings of a Graph
A facial -complete -coloring of a triangulation on a surface is a vertex -coloring such that every triple of -colors appears on the boundary of some face of . The facial -achromatic number of is the maximum integer such that has a facial -complete -coloring. This notion is an expansion of the complete coloring, that is, a proper vertex coloring of a graph such that every pair of colors appears on the ends of some edge.
For two triangulations and on a surface, may not be equal to even if is isomorphic to as graphs. Hence, it would be interesting to see how large the difference between and can be. We shall show that an upper bound for such difference in terms of the genus of the surface
Blocking nonorientability of a surface
AbstractLet S be a nonorientable surface. A collection of pairwise noncrossing simple closed curves in S is a blockage if every one-sided simple closed curve in S crosses at least one of them. Robertson and Thomas [9] conjectured that the orientable genus of any graph G embedded in S with sufficiently large face-width is “roughly” equal to one-half of the minimum number of intersections of a blockage with the graph. The conjecture was disproved by Mohar (Discrete Math. 182 (1998) 245) and replaced by a similar one. In this paper, it is proved that the conjectures in Mohar (1998) and Robertson and Thomas (J. Graph Theory 15 (1991) 407) hold up to a constant error term: For any graph G embedded in S, the orientable genus of G differs from the conjectured value at most by O(g2), where g is the genus of S
On the genera of polyhedral embeddings of cubic graph
In this article we present theoretical and computational results on the
existence of polyhedral embeddings of graphs. The emphasis is on cubic graphs.
We also describe an efficient algorithm to compute all polyhedral embeddings of
a given cubic graph and constructions for cubic graphs with some special
properties of their polyhedral embeddings. Some key results are that even cubic
graphs with a polyhedral embedding on the torus can also have polyhedral
embeddings in arbitrarily high genus, in fact in a genus {\em close} to the
theoretical maximum for that number of vertices, and that there is no bound on
the number of genera in which a cubic graph can have a polyhedral embedding.
While these results suggest a large variety of polyhedral embeddings,
computations for up to 28 vertices suggest that by far most of the cubic graphs
do not have a polyhedral embedding in any genus and that the ratio of these
graphs is increasing with the number of vertices.Comment: The C-program implementing the algorithm described in this article
can be obtained from any of the author
Constant-factor approximations for asymmetric TSP on nearly-embeddable graphs
In the Asymmetric Traveling Salesperson Problem (ATSP) the goal is to find a
closed walk of minimum cost in a directed graph visiting every vertex. We
consider the approximability of ATSP on topologically restricted graphs. It has
been shown by [Oveis Gharan and Saberi 2011] that there exists polynomial-time
constant-factor approximations on planar graphs and more generally graphs of
constant orientable genus. This result was extended to non-orientable genus by
[Erickson and Sidiropoulos 2014].
We show that for any class of \emph{nearly-embeddable} graphs, ATSP admits a
polynomial-time constant-factor approximation. More precisely, we show that for
any fixed , there exist , such that ATSP on
-vertex -nearly-embeddable graphs admits a -approximation in time
. The class of -nearly-embeddable graphs contains graphs with at
most apices, vortices of width at most , and an underlying surface
of either orientable or non-orientable genus at most . Prior to our work,
even the case of graphs with a single apex was open. Our algorithm combines
tools from rounding the Held-Karp LP via thin trees with dynamic programming.
We complement our upper bounds by showing that solving ATSP exactly on graphs
of pathwidth (and hence on -nearly embeddable graphs) requires time
, assuming the Exponential-Time Hypothesis (ETH). This is
surprising in light of the fact that both TSP on undirected graphs and Minimum
Cost Hamiltonian Cycle on directed graphs are FPT parameterized by treewidth