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The structure of totally critical graphs of odd order

2003
*
Discrete Mathematics
*

We show that such

doi:10.1016/s0012-365x(02)00810-5
fatcat:2nis3aussfcwbfnkdalqgiyrdm
*graphs*are*totally**critical*if and only if they are conformability*critical*. We also give a*structural*characterization*of*such*totally**critical**graphs*. ... ,*the*situation for*odd**order**graphs*has been unclear. ... Acknowledgements We would like to thank one*of**the*referees in particular for his very careful reading*of**the*manuscript and his many thoughtful comments which we have tried to take fully into account ...##
###
On total covers of graphs

1992
*
Discrete Mathematics
*

We prove that such a

doi:10.1016/0012-365x(92)90643-t
fatcat:nbkmdu3mkrbjzmtbhyqt7w4er4
*graph*with even*order*has a l-factor and such a*graph*with*odd**order*is factor-*critical*. ... A*total*cover*of*a*graph*G is a subset*of*V(G) UE(G) which covers all elements*of*V(G) U E(G).*The**total*covering number q(G)*of*a*graph*G is*the*minimum cardinality*of*a*total*cover in G. ... Acknowledgments*The*authors wish to thank*the*referees for pointing out*the*result in [3] and other helpful suggestions. ...##
###
Totally Critical Even Order Graphs

1999
*
Journal of combinatorial theory. Series B (Print)
*

A

doi:10.1006/jctb.1999.1902
fatcat:hk6igpj34bfqzcw27ygjxe6sne
*graph*is*totally**critical*if it is Type 2, connected, and*the*removal*of*any edge reduces*the**total*chromatic number. ... A good characterization*of*all*totally**critical**graphs*is unlikely as Sanchez-Arroyo showed that determining*the**total*chromatic number*of*a*graph*is an NP-hard problem. ... We thank one*of**the*referees for some thoughtful advice concerning*the*presentation*of*our results. Also we regret to announce that*the*first author is now deceased. ...##
###
Page 2449 of Mathematical Reviews Vol. , Issue 2000d
[page]

2000
*
Mathematical Reviews
*

In this paper,

*the*authors mainly characterize*the**totally**critical**graphs*G*of*even*order*with A(G) > +|V(G)|. ... [Chew, Kian Hoe] (5-NSW-SM; Sydney)*Total*chromatic number*of**graphs**of**odd**order*and high degree. (English summary) Discrete Math. 205 (1999), no. 1-3, 39-46. ...##
###
Page 3571 of Mathematical Reviews Vol. , Issue 2004e
[page]

2004
*
Mathematical Reviews
*

We show that such

*graphs*are*totally**critical*if and only if they are conformability*critical*. We also give a*structural*characterization*of*such*totally**critical**graphs*. ... contrast,*the*situation for*odd**order**graphs*has been unclear. ...##
###
Weighted graphs defining facets: A connection between stable set and linear ordering polytopes

2009
*
Discrete Optimization
*

*The*class

*of*alpha-

*critical*

*graphs*has several nice

*structural*properties, most

*of*them related to their defect which is

*the*number

*of*vertices minus two times

*the*stability number. ...

*The*class

*of*alpha-

*critical*

*graphs*is also

*of*interest for at least two topics

*of*polyhedral studies. ... Let G H whenever H is an

*odd*subdivision

*of*G. This defines a partial

*order*on

*graphs*. Consider

*the*set

*of*all connected α-

*critical*

*graphs*partially

*ordered*by . ...

##
###
Primitive nonclassical structures of theN-qubit Pauli group

2014
*
Physical Review A. Atomic, Molecular, and Optical Physics
*

We define

doi:10.1103/physreva.89.012321
fatcat:yhho67ha4rc7rclnt7y2oncqty
*the**criticality*, or irreducibility,*of*these*structures*, and its connection to entanglement. ...*The**critical*IDs for a given N are a finite set*of*geometric objects that appear to fully characterize*the*nonclassicality*of**the*N-qubit Pauli group. ... Aravind for many useful discussions as I pursued this research, and he was directly involved in*the*various precursor projects to which I have referred throughout. ...##
###
Recognizing a totally oddK4-subdivision, parity 2-disjoint rooted paths and a parity cycle through specified elements (extended abstract)
[chapter]

2010
*
Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms
*

*The*recognition

*of*a

*totally*

*odd*K4-subdivision plays an important role in both

*graph*theory and combinatorial optimization. ... A

*totally*

*odd*K4-subdivision is a subdivision

*of*K4 where each subdivided edge has

*odd*length. ...

*Totally*

*odd*-K 4 -subdivision Again, we apply

*the*Main

*Structural*Result. ...

##
###
Totally odd subdivisions and parity subdivisions: Structures and Coloring
[chapter]

2013
*
Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms
*

In this paper, we give a

doi:10.1137/1.9781611973105.73
dblp:conf/soda/Kawarabayashi13a
fatcat:p3zj4omjxfg7nbpiepwiiaqqom
*structure*theorem for*graphs*without a fixed*graph*H as a*totally**odd*subdivision. ... A*totally**odd*H-subdivision means a subdivision*of*a*graph*H in which each edge*of*H corresponds to a path*of**odd*length. Thus this concept is a generalization*of*a subdivision*of*H. ...*The*same conclusion*of**the**structure*theorem is true if we replace "*totally**odd*" by "parity". Hence this generalizes*the**structure*theorem for subdivision-free*graphs*[17, 29] . ...##
###
Graph-theoretic Models of Dispositional Structures

2013
*
International Studies in the Philosophy of Science
*

Acknowledgements An early version

doi:10.1080/02698595.2013.783979
fatcat:i64liu5pejch7mn6k5orl7urmq
*of*this paper was presented in Melbourne at*the*metaphysics*of*science conference organized by Brian Ellis and Howard Sankey, and in Nottingham for*the*Royal Institute ...*of*Philosophy Seminar organized by Uri Leibowitz. ... In*order*to save*the**graph*constraints, one could take*the*bold step*of*accepting that a world containing an*odd*number*of*properties in*total*(what I will call an '*odd*' world) simply is impossible. ...##
###
Empirical signatures of shape phase transitions in nuclei with odd nucleon numbers

2018
*
Physical Review C
*

Candidates for

doi:10.1103/physrevc.98.024301
fatcat:e33im3p4dnff7cpagmtuwle6cu
*critical*point nuclei are proposed for*odd*-mass and*odd*-*odd*nuclei, using correlations between relative excitation energies, and their ratios, for*structures*(bands) based on unique-parity ... Nuclear level density at low excitation energies is proposed as an indicator*of**the*first*order*phase transitions in nuclei. ... Acknowledgments Partial funding by*the*Romanian project IFA -CERN-RO/ISOLDE is acknowledged. ...##
###
Page 2923 of Mathematical Reviews Vol. , Issue 80H
[page]

1980
*
Mathematical Reviews
*

(ii) If G is a 3-chromatic

*critical**graph**of*smallest even*order*, then G has no multiple edges and no triangles. ...*The*conjecture states that if G is a*graph*that is*critical*with respect to edge coloring, i.e.,*the*removal*of*any vertex re- duces*the*edge chromatic index, then G has an*odd*number*of*edges. ...##
###
Page 1493 of Mathematical Reviews Vol. 53, Issue 5
[page]

1977
*
Mathematical Reviews
*

Using

*the*concept*of*“*critical*components”*the*author intro- duces a reduction operation by means*of*which*the**structure**of*«-coverable*graphs*can be investigated. ... Let u=(x, y) and v=(y, z) be a pair*of*adjacent*critical*edges*of*a*graph*G without an*odd*hole. Then*the*author proves that G contains*the*edge w=(x, z), too. Some consequences are also discussed. ...##
###
Page 4555 of Mathematical Reviews Vol. , Issue 96h
[page]

1996
*
Mathematical Reviews
*

In

*the*first part*of*this paper we give a uniform way*of*defining*the*line, #(D), middle, ° AD),*total*, 7 (D), and subdivision, ~(D), digraph*of*a di-*graph*D as intersection digraphs*of*families*of**ordered*... They also showed that any*graph*consisting*of*an*odd*cycle with or without isolated and/or pendant vertices is*the*domination*graph**of*some tournament. ...##
###
Page 1331 of Mathematical Reviews Vol. , Issue 96c
[page]

1996
*
Mathematical Reviews
*

Second,

*the*authors give an easier proof to*the*theorem by Seb6 on bi-*critical**graphs*. ... A T-join is a subset J*of*edges*of*a*graph*wherein*the*number*of*edges incident with a vertex v*of*T is*odd*only when v is in T. ...
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