6,807 research outputs found
Even Orientations and Pfaffian graphs
We give a characterization of Pfaffian graphs in terms of even orientations,
extending the characterization of near bipartite non--pfaffian graphs by
Fischer and Little \cite{FL}. Our graph theoretical characterization is
equivalent to the one proved by Little in \cite{L73} (cf. \cite{LR}) using
linear algebra arguments
Ramsey expansions of metrically homogeneous graphs
We discuss the Ramsey property, the existence of a stationary independence
relation and the coherent extension property for partial isometries (coherent
EPPA) for all classes of metrically homogeneous graphs from Cherlin's
catalogue, which is conjectured to include all such structures. We show that,
with the exception of tree-like graphs, all metric spaces in the catalogue have
precompact Ramsey expansions (or lifts) with the expansion property. With two
exceptions we can also characterise the existence of a stationary independence
relation and the coherent EPPA.
Our results can be seen as a new contribution to Ne\v{s}et\v{r}il's
classification programme of Ramsey classes and as empirical evidence of the
recent convergence in techniques employed to establish the Ramsey property, the
expansion (or lift or ordering) property, EPPA and the existence of a
stationary independence relation. At the heart of our proof is a canonical way
of completing edge-labelled graphs to metric spaces in Cherlin's classes. The
existence of such a "completion algorithm" then allows us to apply several
strong results in the areas that imply EPPA and respectively the Ramsey
property.
The main results have numerous corollaries on the automorphism groups of the
Fra\"iss\'e limits of the classes, such as amenability, unique ergodicity,
existence of universal minimal flows, ample generics, small index property,
21-Bergman property and Serre's property (FA).Comment: 57 pages, 14 figures. Extends results of arXiv:1706.00295. Minor
revisio
Zero Forcing Sets and Bipartite Circulants
In this paper we introduce a class of regular bipartite graphs whose
biadjacency matrices are circulant matrices and we describe some of their
properties. Notably, we compute upper and lower bounds for the zero forcing
number for such a graph based only on the parameters that describe its
biadjacency matrix. The main results of the paper characterize the bipartite
circulant graphs that achieve equality in the lower bound.Comment: 22 pages, 13 figure
Conflict-Free Coloring of Planar Graphs
A conflict-free k-coloring of a graph assigns one of k different colors to
some of the vertices such that, for every vertex v, there is a color that is
assigned to exactly one vertex among v and v's neighbors. Such colorings have
applications in wireless networking, robotics, and geometry, and are
well-studied in graph theory. Here we study the natural problem of the
conflict-free chromatic number chi_CF(G) (the smallest k for which
conflict-free k-colorings exist). We provide results both for closed
neighborhoods N[v], for which a vertex v is a member of its neighborhood, and
for open neighborhoods N(v), for which vertex v is not a member of its
neighborhood.
For closed neighborhoods, we prove the conflict-free variant of the famous
Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a
minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case
bound: three colors are sometimes necessary and always sufficient. We also give
a complete characterization of the computational complexity of conflict-free
coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G,
but polynomial for outerplanar graphs. Furthermore, deciding whether
chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for
outerplanar graphs. For the bicriteria problem of minimizing the number of
colored vertices subject to a given bound k on the number of colors, we give a
full algorithmic characterization in terms of complexity and approximation for
outerplanar and planar graphs.
For open neighborhoods, we show that every planar bipartite graph has a
conflict-free coloring with at most four colors; on the other hand, we prove
that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite
graph has a conflict-free k-coloring. Moreover, we establish that any general}
planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on
Discrete Mathematics) of extended abstract that appears in Proceeedings of
the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA
2017), pp. 1951-196
Approximating the Regular Graphic TSP in near linear time
We present a randomized approximation algorithm for computing traveling
salesperson tours in undirected regular graphs. Given an -vertex,
-regular graph, the algorithm computes a tour of length at most
, with high probability, in time. This improves upon a recent result by Vishnoi (\cite{Vishnoi12}, FOCS
2012) for the same problem, in terms of both approximation factor, and running
time. The key ingredient of our algorithm is a technique that uses
edge-coloring algorithms to sample a cycle cover with cycles with
high probability, in near linear time.
Additionally, we also give a deterministic
factor approximation algorithm
running in time .Comment: 12 page
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