12,254 research outputs found
Konstruksi Famili Graf Hampir Planar dengan Angka Perpotongan Tertentu
KONSTRUKSI FAMILI GRAF HAMPIR PLANAR DENGAN ANGKA PERPOTONGAN TERTENTU Benny Pinontoan1) 1) Program Studi Matematika FMIPA Universitas Sam Ratulangi Manado, 95115ABSTRAK Sebuah graf adalah pasangan himpunan tak kosong simpul dan himpunan sisi. Graf dapat digambar pada bidang dengan atau tanpa perpotongan. Angka perpotongan adalah jumlah perpotongan terkecil di antara semua gambar graf pada bidang. Graf dengan angka perpotongan nol disebut planar. Graf memiliki penerapan penting pada desain Very Large Scale of Integration (VLSI). Sebuah graf dinamakan perpotongan kritis jika penghapusan sebuah sisi manapun menurunkan angka perpotongannya, sedangkan sebuah graf dinamakan hampir planar jika menghapus salah satu sisinya membuat graf yang sisa menjadi planar. Banyak famili graf perpotongan kritis yang dapat dibentuk dari bagian-bagian kecil yang disebut ubin yang diperkenalkan oleh Pinontoan dan Richter (2003). Pada tahun 2010, Bokal memperkenalkan operasi perkalian zip untuk graf. Dalam artikel ini ditunjukkan sebuah konstruksi dengan menggunakan ubin dan perkalian zip yang jika diberikan bilangan bulat k ³ 1, dapat menghasilkan famili tak hingga graf hampir planar dengan angka perpotongan k. Kata kunci: angka perpotongan, ubin graf, graf hampir planar. CONSTRUCTION OF INFINITE FAMILIES OF ALMOST PLANAR GRAPH WITH GIVEN CROSSING NUMBER ABSTRACT A graph is a pair of a non-empty set of vertices and a set of edges. Graphs can be drawn on the plane with or without crossing of its edges. Crossing number of a graph is the minimal number of crossings among all drawings of the graph on the plane. Graphs with crossing number zero are called planar. Crossing number problems find important applications in the design of layout of Very Large Scale of Integration (VLSI). A graph is crossing-critical if deleting of any of its edge decreases its crossing number. A graph is called almost planar if deleting one edge makes the graph planar. Many infinite sequences of crossing-critical graphs can be made up by gluing small pieces, called tiles introduced by Pinontoan and Richter (2003). In 2010, Bokal introduced the operation zip product of graphs. This paper shows a construction by using tiles and zip product, given an integer k ³ 1, to build an infinite family of almost planar graphs having crossing number k
Nested cycles in large triangulations and crossing-critical graphs
We show that every sufficiently large plane triangulation has a large
collection of nested cycles that either are pairwise disjoint, or pairwise
intersect in exactly one vertex, or pairwise intersect in exactly two vertices.
We apply this result to show that for each fixed positive integer , there
are only finitely many -crossing-critical simple graphs of average degree at
least six. Combined with the recent constructions of crossing-critical graphs
given by Bokal, this settles the question of for which numbers there is
an infinite family of -crossing-critical simple graphs of average degree
Exact Potts Model Partition Functions on Strips of the Honeycomb Lattice
We present exact calculations of the partition function of the -state
Potts model on (i) open, (ii) cyclic, and (iii) M\"obius strips of the
honeycomb (brick) lattice of width and arbitrarily great length. In the
infinite-length limit the thermodynamic properties are discussed. The
continuous locus of singularities of the free energy is determined in the
plane for fixed temperature and in the complex temperature plane for fixed
values. We also give exact calculations of the zero-temperature partition
function (chromatic polynomial) and , the exponent of the ground-state
entropy, for the Potts antiferromagnet for honeycomb strips of type (iv)
, cyclic, (v) , M\"obius, (vi) , cylindrical, and (vii)
, open. In the infinite-length limit we calculate and determine
the continuous locus of points where it is nonanalytic. We show that our exact
calculation of the entropy for the strip with cylindrical boundary
conditions provides an extremely accurate approximation, to a few parts in
for moderate values, to the entropy for the full 2D honeycomb
lattice (where the latter is determined by Monte Carlo measurements since no
exact analytic form is known).Comment: 48 pages, latex, with encapsulated postscript figure
Bond percolation on isoradial graphs: criticality and universality
In an investigation of percolation on isoradial graphs, we prove the
criticality of canonical bond percolation on isoradial embeddings of planar
graphs, thus extending celebrated earlier results for homogeneous and
inhomogeneous square, triangular, and other lattices. This is achieved via the
star-triangle transformation, by transporting the box-crossing property across
the family of isoradial graphs. As a consequence, we obtain the universality of
these models at the critical point, in the sense that the one-arm and
2j-alternating-arm critical exponents (and therefore also the connectivity and
volume exponents) are constant across the family of such percolation processes.
The isoradial graphs in question are those that satisfy certain weak conditions
on their embedding and on their track system. This class of graphs includes,
for example, isoradial embeddings of periodic graphs, and graphs derived from
rhombic Penrose tilings.Comment: In v2: extended title, and small changes in the tex
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