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 $k$, there are only finitely many $k$-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 $q>0$ there is an infinite family of $k$-crossing-critical simple graphs of average degree $q$

Duality and free energy analyticity bounds for few-body Ising models with extensive homology rank

We consider pairs of few-body Ising models where each spin enters a bounded number of interaction terms (bonds) such that each model can be obtained from the dual of the other after freezing k spins on large-degree sites. Such a pair of Ising models can be interpreted as a two-chain complex with k being the rank of the first homology group. Our focus is on the case where k is extensive, that is, scales linearly with the number of bonds n. Flipping any of these additional spins introduces a homologically nontrivial defect (generalized domain wall). In the presence of bond disorder, we prove the existence of a low-temperature weak-disorder region where additional summation over the defects has no effect on the free energy density f(T) in the thermodynamical limit and of a high-temperature region where an extensive homological defect does not affect f(T). We also discuss the convergence of the high- and low-temperature series for the free energy density, prove the analyticity of limiting f(T) at high and low temperatures, and construct inequalities for the critical point(s) where analyticity is lost. As an application, we prove multiplicity of the conventionally defined critical points for Ising models on all { f, d} tilings of the infinite hyperbolic plane, where df/(d + f) \u3e 2. Namely, for these infinite graphs, we show that critical temperatures with free and wired boundary conditions differ, Tc(f)T(f)

Zeros of the Potts Model Partition Function on Sierpinski Graphs

We calculate zeros of the $q$-state Potts model partition function on $m$'th-iterate Sierpinski graphs, $S_m$, in the variable $q$ and in a temperature-like variable, $y$. We infer some asymptotic properties of the loci of zeros in the limit $m \to \infty$ and relate these to thermodynamic properties of the $q$-state Potts ferromagnet and antiferromagnet on the Sierpinski gasket fractal, $S_\infty$.Comment: 6 pages, 8 figure

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