Smallest $$C_{2l+1}$$-Critical Graphs of Odd-Girth $$2k+1$$

2020 ◽  
pp. 184-196
Author(s):  
Laurent Beaudou ◽  
Florent Foucaud ◽  
Reza Naserasr
Keyword(s):  
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 Counting Critical Subgraphs in k-Critical Graphs

COMBINATORICA ◽  
2021 ◽  
Author(s):  
Jie Ma ◽  
Tianchi Yang
Keyword(s):  
Critical Graphs

Excluding induced subgraphs: Critical graphs

2010 ◽  
Vol 38 (1-2) ◽  
pp. 100-120 ◽  
Author(s):  
József Balogh ◽  
Jane Butterfield
Keyword(s):  
Induced Subgraphs ◽  
Critical Graphs

scholarly journals Subcubic Edge-Chromatic Critical Graphs Have Many Edges

2017 ◽  
Vol 86 (1) ◽  
pp. 122-136 ◽  
Author(s):  
Daniel W. Cranston ◽  
Landon Rabern
Keyword(s):  
Critical Graphs

scholarly journals On the existence problem of the total domination vertex critical graphs

2011 ◽  
Vol 159 (1) ◽  
pp. 46-52 ◽  
Author(s):  
Moo Young Sohn ◽  
Dongseok Kim ◽  
Young Soo Kwon ◽  
Jaeun Lee
Keyword(s):  
Existence Problem ◽  
Total Domination ◽  
Critical Graphs

On all fractional (a, b, k)-critical graphs

2014 ◽  
Vol 30 (4) ◽  
pp. 696-702 ◽  
Author(s):  
Si Zhong Zhou ◽  
Zhi Ren Sun
Keyword(s):  
Critical Graphs

k-Extendable Graphs and n-Factor-Critical Graphs

2009 ◽  
pp. 209-249 ◽  
Author(s):  
Qinglin Roger Yu ◽  
Guizhen Liu
Keyword(s):  
Critical Graphs

scholarly journals Spectral strengthening of a theorem on transversal critical graphs

2022 ◽  
Vol 345 (3) ◽  
pp. 112717
Author(s):  
Muhuo Liu ◽  
Xiaofeng Gu
Keyword(s):  
Critical Graphs

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 $k$.

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