Ear Decomposition of Factor-Critical Graphs and Number of Maximum Matchings

2015 ◽  
Vol 38 (4) ◽  
pp. 1537-1549 ◽  
Author(s):  
Yan Liu ◽  
Shenlin Zhang
Keyword(s):  
Critical Graphs ◽  
Ear Decomposition

scholarly journals Connected τ -critical hypergraphs of minimal size

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Matěj Stehlík
Keyword(s):  
Classical Theorem ◽  
Minimal Size ◽  
Critical Graphs ◽  
International Audience ◽  
Ear Decomposition ◽  
Transversal Number

International audience A hypergraph $\mathscr{H}$ is $τ$ -critical if $τ (\mathscr{H}-E) < τ (\mathscr{H})$ for every edge $E ∈\mathscr{H}$, where $τ (\mathscr{H})$ denotes the transversal number of $\mathscr{H}$. It can be shown that a connected $τ$ -critical hypergraph $\mathscr{H}$ has at least $2τ (\mathscr{H})-1$ edges; this generalises a classical theorem of Gallai on $χ$ -vertex-critical graphs with connected complements. In this paper we study connected $τ$ -critical hypergraphs $\mathscr{H}$ with exactly $2τ (\mathscr{H)}-1$ edges. We prove that such hypergraphs have at least $2τ (\mathscr{H})-1$ vertices, and characterise those with $2τ (\mathscr{H})-1$ vertices using a directed odd ear decomposition of an associated digraph. Using Seymour's characterisation of $χ$ -critical 3-chromatic square hypergraphs, we also show that a connected square hypergraph $\mathscr{H}$ with fewer than $2τ (\mathscr{H})$ edges is $τ$ -critical if and only if it is $χ$ -critical 3-chromatic. Finally, we deduce some new results on $χ$ -vertex-critical graphs with connected complements.

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

Ear decomposition of 3-regular polyhedral links with applications

2014 ◽  
Vol 359 ◽  
pp. 146-154 ◽  
Author(s):  
Xiao-Sheng Cheng ◽  
Heping Zhang ◽  
Xian׳an Jin ◽  
Wen-Yuan Qiu
Keyword(s):  
Polyhedral Links ◽  
Ear Decomposition

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
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