Infinite families of crossing-critical graphs with prescribed average degree and crossing number

2010 ◽  
Vol 65 (2) ◽  
pp. 139-162 ◽  
Author(s):  
Drago Bokal
Keyword(s):  
Crossing Number ◽  
Average Degree ◽  
Critical Graphs

scholarly journals Improvement on the Crossing Number of Crossing-Critical Graphs

2020 ◽  
Author(s):  
János Barát ◽  
Géza Tóth
Keyword(s):  
Crossing Number ◽  
Critical Graphs ◽  
Minimum Number ◽  
Edge Crossings

AbstractThe crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. A graph G is k-crossing-critical if its crossing number is at least k, but if we remove any edge of G, its crossing number drops below k. There are examples of k-crossing-critical graphs that do not have drawings with exactly k crossings. Richter and Thomassen proved in 1993 that if G is k-crossing-critical, then its crossing number is at most $$2.5\, k+16$$ 2.5 k + 16 . We improve this bound to $$2k+8\sqrt{k}+47$$ 2 k + 8 k + 47 .

scholarly journals On Degree Properties of Crossing-Critical Families of Graphs

10.37236/7753 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Drago Bokal ◽  
Mojca Bračič ◽  
Marek Derňár ◽  
Petr Hliněný
Keyword(s):  
Average Degree ◽  
Critical Graphs ◽  
Odd Degree ◽  
Vertex Degrees ◽  
Open Question

Answering an open question from 2007, we construct infinite $k$-crossing-critical families of graphs that contain vertices of any prescribed odd degree, for any sufficiently large $k$. To answer this question, we introduce several properties of infinite families of graphs and operations on the families allowing us to obtain new families preserving those properties. This conceptual setup allows us to answer general questions on behaviour of degrees in crossing-critical graphs: we show that, for any set of integers $D$ such that $\min(D)\geq 3$ and $3,4\in D$, and for any sufficiently large $k$, there exists a $k$-crossing-critical family such that the numbers in $D$ are precisely the vertex degrees that occur arbitrarily often in (large enough) graphs of this family. Furthermore, even if both $D$ and some average degree in the interval $(3,6)$ are prescribed, $k$-crossing-critical families exist for any sufficiently large $k$.

scholarly journals On the average degree of critical graphs with maximum degree six

2011 ◽  
Vol 311 (21) ◽  
pp. 2574-2576 ◽  
Author(s):  
Lianying Miao ◽  
Jibin Qu ◽  
Qingbo Sun
Keyword(s):  
Maximum Degree ◽  
Average Degree ◽  
Critical Graphs

scholarly journals A Better Lower Bound on Average Degree of Online $k$-List-Critical Graphs

10.37236/6405 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Landon Rabern
Keyword(s):  
Lower Bound ◽  
Average Degree ◽  
Critical Graphs ◽  
Critical Graph

We improve the best known bounds on average degree of online $k$-list-critical graphs for $k \geqslant 6$. Specifically, for $k \geqslant 7$ we show that every non-complete online $k$-list-critical graph has average degree at least $k-1 + \frac{(k-3)^2 (2 k-3)}{k^4-2 k^3-11 k^2+28 k-14}$ and every non-complete online $6$-list-critical graph has average degree at least $5 + \frac{93}{766}$. The same bounds hold for offline $k$-list-critical graphs.

scholarly journals A Better Lower Bound on Average Degree of 4-List-Critical Graphs

10.37236/5971 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Landon Rabern
Keyword(s):  
Lower Bound ◽  
Short Note ◽  
Average Degree ◽  
Critical Graphs ◽  
Critical Graph

This short note proves that every non-complete $k$-list-critical graph has average degree at least $k-1 + \frac{k-3}{k^2-2k+2}$. This improves the best known bound for $k = 4,5,6$. The same bound holds for online $k$-list-critical graphs.

scholarly journals New Infinite Families of Almost-Planar Crossing-Critical Graphs

10.37236/826 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Petr Hliněný
Keyword(s):  
Average Degree ◽  
The Other ◽  
Critical Graphs

We show that, for all choices of integers $k>2$ and $m$, there are simple $3$-connected $k$-crossing-critical graphs containing more than $m$ vertices of each even degree $\leq2k-2$. This construction answers one half of a question raised by Bokal, while the other half asking analogously about vertices of odd degrees at least $7$ in crossing-critical graphs remains open. Furthermore, our newly constructed graphs have several other interesting properties; for instance, they are almost planar and their average degree can attain any rational value in the interval $\big[3+{1\over5},6-{8\over k+1}\big)$.

scholarly journals Book Embedding of Infinite Family ((2h+3 2))-Crossing-Critical Graphs for h=1 with Rational Average Degree r∈(3.5,4)

d'CARTESIAN ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 145
Author(s):  
Sheren H. Wilar ◽  
Benny Pinontoan ◽  
Chriestie E.J.C. Montolalu
Keyword(s):  
Intersection Number ◽  
Infinite Family ◽  
Infinite Graph ◽  
Average Degree ◽  
Critical Graphs ◽  
Book Embedding ◽  
Embed Graph ◽  
Principal Tool ◽  
Page Book

A principal tool used in construction of crossing-critical graphs are tiles. In the tile concept, tiles can be arranged by gluing one tile to another in a linear or circular fashion. The series of tiles with circular fashion form an infinite graph family. In this way, the intersection number of this family of graphs can be determined. In this research, has been formed an infinite family graphs Q_((1,s,b) ) (n) with average degree r between 3.5 and 4. The graph formed by gluing together many copies of the tile P_((1,s,b) ) in circular fashion, where the tile P_((1,s,b) ) consist of two identical pieces of tile. And then, the graph embedded into the book to determine the pagenumber that can be formed. When embed graph into book, the vertices are put on a line called the spine and the edges are put on half-planes called the pages. The results obtained show that the graph Q_((1,s,b) ) (n) has 10-crossing-critical and book embedding of graph has 4-page book.

scholarly journals KONSTRUKSI FAMILI GRAF HAMPIR PLANAR DENGAN ANGKA PERPOTONGAN TERTENTU

2011 ◽  
Vol 15 (1) ◽  
pp. 150
Author(s):  
Benny Pinontoan
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Large Scale ◽  
Infinite Family ◽  
Crossing Number ◽  
Plane Graphs ◽  
Critical Graphs ◽  
Product Of Graphs ◽  
Infinite Sequences

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. Keywords: Crossing number, tile, almost planar graph.

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