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 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 $k$.

One-Page Book Embedding under Vertex-Neighborhood Constraints

1990 ◽  
Vol 3 (3) ◽  
pp. 376-390 ◽  
Author(s):  
Shlomo Moran ◽  
Yaron Wolfstahl
Keyword(s):  
Book Embedding ◽  
Page Book

An Infinite Family of Sum-Paint Critical Graphs

2020 ◽  
Vol 36 (5) ◽  
pp. 1563-1571
Author(s):  
Thomas Mahoney ◽  
Chad Wiley
Keyword(s):  
Infinite Family ◽  
Critical Graphs

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 An advance in infinite graph models for the analysis of transportation networks

2016 ◽  
Vol 26 (4) ◽  
pp. 855-869
Author(s):  
Martín Cera ◽  
Eugenio M. Fedriani
Keyword(s):  
Graph Theory ◽  
Transportation Networks ◽  
Finite Graph ◽  
Infinite Graph ◽  
Average Degree ◽  
Infinite Graphs ◽  
Graph Models ◽  
Dangerous Goods ◽  
Finite Case ◽  
Percolation Thresholds

Abstract This paper extends to infinite graphs the most general extremal issues, which are problems of determining the maximum number of edges of a graph not containing a given subgraph. It also relates the new results with the corresponding situations for the finite case. In particular, concepts from ‘finite’ graph theory, like the average degree and the extremal number, are generalized and computed for some specific cases. Finally, some applications of infinite graphs to the transportation of dangerous goods are presented; they involve the analysis of networks and percolation thresholds.

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