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The structure of totally critical graphs of odd order

J.K Dugdale, A.J.W Hilton
2003 Discrete Mathematics  
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.  ...  , 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  ... 
doi:10.1016/s0012-365x(02)00810-5 fatcat:2nis3aussfcwbfnkdalqgiyrdm

On total covers of graphs

Yousef Alavi, Jiuqiang Liu, Jianfang Wang, Zhongfu Zhang
1992 Discrete Mathematics  
We prove that such a 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.  ... 
doi:10.1016/0012-365x(92)90643-t fatcat:nbkmdu3mkrbjzmtbhyqt7w4er4

Totally Critical Even Order Graphs

G.M. Hamilton, A.J.W. Hilton, H.R.F. Hind
1999 Journal of combinatorial theory. Series B (Print)  
A 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.  ... 
doi:10.1006/jctb.1999.1902 fatcat:hk6igpj34bfqzcw27ygjxe6sne

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

Jean-Paul Doignon, Samuel Fiorini, Gwenaël Joret
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 .  ... 
doi:10.1016/j.disopt.2008.07.001 fatcat:5nuwlkhw55gfjeb7apxhxipqd4

Primitive nonclassical structures of theN-qubit Pauli group

Mordecai Waegell
2014 Physical Review A. Atomic, Molecular, and Optical Physics  
We define 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.  ... 
doi:10.1103/physreva.89.012321 fatcat:yhho67ha4rc7rclnt7y2oncqty

Recognizing a totally oddK4-subdivision, parity 2-disjoint rooted paths and a parity cycle through specified elements (extended abstract) [chapter]

Ken-ichi Kawarabayashi, Zhentao Li, Bruce Reed
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.  ... 
doi:10.1137/1.9781611973075.27 dblp:conf/soda/KawarabayashiLR10 fatcat:4caqfa24ozfxzlq3kmbelh4jo4

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

Ken-ichi Kawarabayashi
2013 Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms  
In this paper, we give a 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] .  ... 
doi:10.1137/1.9781611973105.73 dblp:conf/soda/Kawarabayashi13a fatcat:p3zj4omjxfg7nbpiepwiiaqqom

Graph-theoretic Models of Dispositional Structures

Matthew Tugby
2013 International Studies in the Philosophy of Science  
Acknowledgements An early version 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.  ... 
doi:10.1080/02698595.2013.783979 fatcat:i64liu5pejch7mn6k5orl7urmq

Empirical signatures of shape phase transitions in nuclei with odd nucleon numbers

D. Bucurescu, N. V. Zamfir
2018 Physical Review C  
Candidates for 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.  ... 
doi:10.1103/physrevc.98.024301 fatcat:e33im3p4dnff7cpagmtuwle6cu

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 ofcritical 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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