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 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 Study on Incompatibility of Traditional Chinese Medicine: Evidence from Formula Network, Chemical Space, and Metabolism Room

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Wei Long ◽  
Xiao-Dong Zhang ◽  
Hong-Ying Wu ◽  
Jin Jin ◽  
Guang-Yun Yu ◽  
...  
Keyword(s):  
Chinese Medicine ◽  
Traditional Chinese Medicine ◽  
Complex Network ◽  
Chemical Space ◽  
Average Distance ◽  
Average Degree ◽  
Clustering Coefficient ◽  
The Other ◽  
Network Diameter ◽  
Lines Of Evidence

A traditional Chinese medicine (TCM) formula network including 362 TCM formulas was built by using complex network methodologies. The properties of this network were analyzed including network diameter, average distance, clustering coefficient, and average degree. Meanwhile, we built a TCM chemical space and a TCM metabolism room under the theory of chemical space. The properties of chemical space and metabolism room were calculated and analyzed. The properties of the medicine pairs in “eighteen antagonisms and nineteen mutual inhibitors,” an ancient rule for TCM incompatibility, were studied based on the TCM formula network, chemical space, and metabolism room. The results showed that the properties of these incompatible medicine pairs are different from those of the other TCM based on the analysis of the TCM formula network, chemical space, and metabolism room. The lines of evidence derived from our work demonstrated that the ancient rule of TCM incompatibility, “eighteen antagonisms and nineteen mutual inhibitors,” is probably scientifically based.

scholarly journals Episteemilise modaalsuse leksikaalsete väljendusvahendite tajumisest: arvatavasti, äkki ja võib-olla

2016 ◽  
Vol 7 (2) ◽  
pp. 187-207
Author(s):  
Laura Tüüts ◽  
Reili Argus
Keyword(s):  
Foreign Language ◽  
Language Skills ◽  
Average Degree ◽  
Epistemic Modality ◽  
The Other ◽  
Epistemic Value ◽  
Context Sentence ◽  
Context Factors ◽  
Degree Of Certainty ◽  
Foreign Language Skills

Kokkuvõte. Artiklis on vaatluse all episteemilise modaalsuse leksikaalsete väljendusvahendite (markerite) arvatavasti, võib-olla ja äkki tõenäosuse tajumine. Vaadeldakse, kui palju oleneb markerite tõenäosuse tajumine kontekstist, näiteks tugevast või nõrgast eeldusest ehk olemasoleva informatsiooni rohkusest, markeriga lause agendi soost ja keelevälistest teguritest ehk vastajate soost ja vanusest. Markerite tõenäosuse taju uurimiseks koostati internetikatse, millele vastas 328 eesti keelt kõnelevat inimest, kellest noorim oli alla 18-aastane ja vanim üle 60. Tulemustest selgus, et kolmest markerist tajutakse kõige tugevamat tõenäosust väljendavana markerit arvatavasti, pisut nõrgemana tunnetatakse võib-olla ja äkki tõenäosust. Kontekstil on markerite tõenäosuse tajumisel oluline roll: tõenäolisemana tajutakse lauseid, mille kontekst on tugevama eeldusega, mitmetähendusliku kontekstiga lauseid tajutakse aga hoopis nõrgema tõenäosusega. Markerite tõenäosuse tajumist mõjutavad ka keelevälised tegurid, nagu vastaja sugu ja vanus.Abstract. Laura Tüüts and Reili Argus: The comprehension of lexical epistemic markers: the Estonian adverbs arvatavasti (‘probably’), äkki (‘perhaps’) and võib-olla (‘maybe’). The paper analyses the comprehension of epistemic modality markers arvatavasti ‘probably’, võib-olla ‘maybe’ and äkki ‘perhaps’. The study focuses on the connections between the nature and the amount of context in the comprehension of epistemic markers which belong to the class of markers with average degree of certainty. In addition to the influence of context, factors like the gender of the agent in the context sentence, as well as the respondent’s gender, age, education and foreign language skills have been analysed according to different markers and context types. Answers of 328 Estonian-speaking respondents (aged 18–60) revealed that the marker arvatavasti ‘probably’ has the strongest epistemic value, while the other two markers under observation have a somewhat weaker epistemic value. Context has a big impact on the comprehension of all three markers: sentences with stronger premises were comprehended as expressing stronger certainty, and the sentences with ambiguous context were comprehended as expressing the weakest certainty. Factors like the gender of the agent in the context sentence as well as the respondent’s gender have some influence on the comprehension of epistemic markers, while factors like foreign language skills and education do not have such a strong effect on the comprehension of epistemic modality markers.Keywords: epistemic modality; lexical markers; comprehension of probability; Estonian

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 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.

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