125 research outputs found
A Decomposition of Gallai Multigraphs
An edge-colored cycle is rainbow if its edges are colored with distinct colors. A Gallai (multi)graph is a simple, complete, edge-colored (multi)graph lacking rainbow triangles. As has been previously shown for Gallai graphs, we show that Gallai multigraphs admit a simple iterative construction. We then use this structure to prove Ramsey-type results within Gallai colorings. Moreover, we show that Gallai multigraphs give rise to a surprising and highly structured decomposition into directed trees
Parameterized Rural Postman Problem
The Directed Rural Postman Problem (DRPP) can be formulated as follows: given
a strongly connected directed multigraph with nonnegative integral
weights on the arcs, a subset of and a nonnegative integer ,
decide whether has a closed directed walk containing every arc of and
of total weight at most . Let be the number of weakly connected
components in the the subgraph of induced by . Sorge et al. (2012) ask
whether the DRPP is fixed-parameter tractable (FPT) when parameterized by ,
i.e., whether there is an algorithm of running time where is a
function of only and the notation suppresses polynomial factors.
Sorge et al. (2012) note that this question is of significant practical
relevance and has been open for more than thirty years. Using an algebraic
approach, we prove that DRPP has a randomized algorithm of running time
when is bounded by a polynomial in the number of vertices in
. We also show that the same result holds for the undirected version of
DRPP, where is a connected undirected multigraph
The graph bottleneck identity
A matrix is said to determine a
\emph{transitional measure} for a digraph on vertices if for all
the \emph{transition inequality} holds and reduces to the equality (called the \emph{graph
bottleneck identity}) if and only if every path in from to contains
. We show that every positive transitional measure produces a distance by
means of a logarithmic transformation. Moreover, the resulting distance
is \emph{graph-geodetic}, that is,
holds if and only if every path in connecting and contains .
Five types of matrices that determine transitional measures for a digraph are
considered, namely, the matrices of path weights, connection reliabilities,
route weights, and the weights of in-forests and out-forests. The results
obtained have undirected counterparts. In [P. Chebotarev, A class of
graph-geodetic distances generalizing the shortest-path and the resistance
distances, Discrete Appl. Math., URL
http://dx.doi.org/10.1016/j.dam.2010.11.017] the present approach is used to
fill the gap between the shortest path distance and the resistance distance.Comment: 12 pages, 18 references. Advances in Applied Mathematic
Recommended from our members
Graph Theory
Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures
Random cubic planar graphs converge to the Brownian sphere
In this paper, the scaling limit of random connected cubic planar graphs (respectively multigraphs) is shown to be the Brownian sphere.
The proof consists in essentially two main steps. First, thanks to the known decomposition of cubic planar graphs into their 3-connected components, the metric structure of a random cubic planar graph is shown to be well approximated by its unique 3-connected component of linear size, with modified distances.
Then, Whitney's theorem ensures that a 3-connected cubic planar graph is the dual of a simple triangulation, for which it is known that the scaling limit is the Brownian sphere. Curien and Le Gall have recently developed a framework to study the modification of distances in general triangulations and in their dual. By extending this framework to simple triangulations, it is shown that 3-connected cubic planar graphs with modified distances converge jointly with their dual triangulation to the Brownian sphere
Random cubic planar graphs converge to the Brownian sphere
In this paper, the scaling limit of random connected cubic planar graphs
(respectively multigraphs) is shown to be the Brownian sphere.
The proof consists in essentially two main steps. First, thanks to the known
decomposition of cubic planar graphs into their 3-connected components, the
metric structure of a random cubic planar graph is shown to be well
approximated by its unique 3-connected component of linear size, with modified
distances.
Then, Whitney's theorem ensures that a 3-connected cubic planar graph is the
dual of a simple triangulation, for which it is known that the scaling limit is
the Brownian sphere. Curien and Le Gall have recently developed a framework to
study the modification of distances in general triangulations and in their
dual. By extending this framework to simple triangulations, it is shown that
3-connected cubic planar graphs with modified distances converge jointly with
their dual triangulation to the Brownian sphere.Comment: 55 page
Weak degeneracy of graphs
Motivated by the study of greedy algorithms for graph coloring, we introduce
a new graph parameter, which we call weak degeneracy. By definition, every
-degenerate graph is also weakly -degenerate. On the other hand, if
is weakly -degenerate, then (and, moreover, the same
bound holds for the list-chromatic and even the DP-chromatic number of ). It
turns out that several upper bounds in graph coloring theory can be phrased in
terms of weak degeneracy. For example, we show that planar graphs are weakly
-degenerate, which implies Thomassen's famous theorem that planar graphs are
-list-colorable. We also prove a version of Brooks's theorem for weak
degeneracy: a connected graph of maximum degree is weakly
-degenerate unless . (By contrast, all -regular
graphs have degeneracy .) We actually prove an even stronger result, namely
that for every , there is such that if is a graph
of weak degeneracy at least , then either contains a -clique or
the maximum average degree of is at least . Finally, we show
that graphs of maximum degree and either of girth at least or of
bounded chromatic number are weakly -degenerate, which
is best possible up to the value of the implied constant.Comment: 21 p
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