Infinite families of crossing-critical graphs with given average degree

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 $k$.

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On the average degree of critical graphs with maximum degree six

Discrete Mathematics ◽

10.1016/j.disc.2011.06.029 ◽

2011 ◽

Vol 311 (21) ◽

pp. 2574-2576 ◽

Cited By ~ 2

Author(s):

Lianying Miao ◽

Jibin Qu ◽

Qingbo Sun

Keyword(s):

Maximum Degree ◽

Average Degree ◽

Critical Graphs

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

The Electronic Journal of Combinatorics ◽

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.

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

The Electronic Journal of Combinatorics ◽

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.

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New Infinite Families of Almost-Planar Crossing-Critical Graphs

The Electronic Journal of Combinatorics ◽

10.37236/826 ◽

2008 ◽

Vol 15 (1) ◽

Cited By ~ 5

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 $\big[3+{1\over5},6-{8\over k+1}\big)$.

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

d'CARTESIAN ◽

10.35799/dc.9.2.2020.29166 ◽

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.

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