scholarly journals 4-Critical Graphs on Surfaces Without Contractible $(\le\!4)$-Cycles

2014 ◽  
Vol 28 (1) ◽  
pp. 521-552 ◽  
Author(s):  
Zdeněk Dvořák ◽  
Bernard Lidický
Keyword(s):  
Critical Graphs ◽  
Graphs On Surfaces

scholarly journals Three-coloring triangle-free graphs on surfaces II. 4-critical graphs in a disk

2018 ◽  
Vol 132 ◽  
pp. 1-46 ◽  
Author(s):  
Zdeněk Dvořák ◽  
Daniel Král' ◽  
Robin Thomas
Keyword(s):  
Critical Graphs ◽  
Graphs On Surfaces

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 .

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
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