32,425 research outputs found
Types of triangle in plane Hamiltonian triangulations and applications to domination and k-walks
We investigate the minimum number t(0)(G) of faces in a Hamiltonian triangulation G so that any Hamiltonian cycle C of G has at least t(0)(G) faces that do not contain an edge of C. We prove upper and lower bounds on the maximum of these numbers for all triangulations with a fixed number of facial triangles. Such triangles play an important role when Hamiltonian cycles in triangulations with 3-cuts are constructed from smaller Hamiltonian cycles of 4-connected subgraphs. We also present results linking the number of these triangles to the length of 3-walks in a class of triangulation and to the domination number
Coloring directed cycles
Sopena in his survey [E. Sopena, The oriented chromatic number of graphs: A
short survey, preprint 2013] writes, without any proof, that an oriented cycle
can be colored with three colors if and only if ,
where is the number of forward arcs minus the number of
backward arcs in . This is not true. In this paper we show that can be colored with three colors if and only if
or does not contain three consecutive arcs going in the same
direction
Generating 5-regular planar graphs
For k = 0, 1, 2, 3, 4, 5, let Pk be the class of k-edge-connected 5-regular planar graphs. In this paper, graph operations are introduced that generate all graphs in each Pk. © 2009 Wiley Periodicals, Inc
The circumference of a graph with no K3,t-minor, II
The class of graphs with no K3;t-minors, t>=3, contains all planar graphs and plays an important role in graph minor theory. In 1992, Seymour and Thomas conjectured the existence of a function α(t)>0 and a constant β>0, such that every 3-connected n-vertex graph with no K3;t-minors, t>=3, contains a cycle of length at least α(t)nβ. The purpose of this paper is to con¯rm this conjecture with α(t)=(1/2)t(t-1) and β=log1729 2.preprin
Flip cycles in plabic graphs
Planar bicolored (plabic) graphs are combinatorial objects introduced by
Postnikov to give parameterizations of the positroid cells of the totally
nonnegative Grassmannian . Any two plabic graphs for
the same positroid cell can be related by a sequence of certain moves. The flip
graph has plabic graphs as vertices and has edges connecting the plabic graphs
which are related by a single move. A recent result of Galashin shows that
plabic graphs can be seen as cross-sections of zonotopal tilings for the cyclic
zonotope . Taking this perspective, we show that the fundamental group
of the flip graph is generated by cycles of length 4, 5, and 10, and use this
result to prove a related conjecture of Dylan Thurston about triple crossing
diagrams. We also apply our result to make progress on an instance of the
generalized Baues problem.Comment: 26 pages, 7 figures. Journal versio
Approximating branchwidth on parametric extensions of planarity
The \textsl{branchwidth} of a graph has been introduced by Roberson and
Seymour as a measure of the tree-decomposability of a graph, alternative to
treewidth. Branchwidth is polynomially computable on planar graphs by the
celebrated ``Ratcatcher''-algorithm of Seymour and Thomas. We investigate an
extension of this algorithm to minor-closed graph classes, further than planar
graphs as follows: Let be a graph embeddedable in the projective plane
and be a graph embeddedable in the torus. We prove that every
-minor free graph contains a subgraph where the
difference between the branchwidth of and the branchwidth of is
bounded by some constant, depending only on and . Moreover, the
graph admits a tree decomposition where all torsos are planar. This
decomposition can be used for deriving an EPTAS for branchwidth: For
-minor free graphs, there is a function
and a -approximation algorithm
for branchwidth, running in time for every
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