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3- and 4-critical graphs of small even order

G. Brinkmann, E. Steffen
1997 Discrete Mathematics  
We show that there is exactly one 3-critical graph of order 22 and that there are exactly nine of order 24. The graph of order 22 is shown to be the smallest 3-critical graph of even order.  ...  We also show that there are exactly two 4-critical graphs of order 18, and that these graphs are the smallest 4-critical graphs of even order. The results are obtained with the aid of a computer.  ...  It is known that there are no critical graphs of even order less than 12, that there are no 3-critical graphs of order 12 or 14 (see [15] ), and that there are no 4-critical graphs of order 12 [3, 5]  ... 
doi:10.1016/s0012-365x(96)00105-7 fatcat:sb2jte6nerbcto4pzs3m7c4lbi

Totally Critical Even Order Graphs

G.M. Hamilton, A.J.W. Hilton, H.R.F. Hind
1999 Journal of combinatorial theory. Series B (Print)  
We also show that if the maximum degree 2 of a non-conformable graph G of even order satisfies 1 2 |V | 2 3 4 |V | &1 then G is regular and has a simple structure. Academic Press  ...  In this paper we show that if the Conformability Conjecture is correct, then totally critical graphs of even order with maximum degree at least one half of their order are characterized by a simple equation  ...  The third author also acknowledges subsequent support received from the Natural Sciences and Engineering Research Council of Canada (Grant OGP 0155408).  ... 
doi:10.1006/jctb.1999.1902 fatcat:hk6igpj34bfqzcw27ygjxe6sne

On small graphs critical with respect to edge colourings

Lowell W. Beineke, Stanley Fiorini
1976 Discrete Mathematics  
Fwrinr / On mall graphs 111 an r-critical gruph of even order n. If n s 10 or if r = 3 and 1 -factor. Theorem 1 is a well-known result on edge-colouri?  ...  The portion of the theorem with r = 3 was proved by Fiorini 131. Therefore we assume that G is an r-critical graph (r Z= 4) of even order n s 10 which has no I-factklr.  ...  (I -1) in H, I' 4: N must h'ive 1st 23" and then ls(GP) is at least 8, which contradrcts the d&iencv condition.  ... 
doi:10.1016/0012-365x(76)90139-4 fatcat:xvh3fk6kpjhjdksmrbowx2giey

Critical concepts in domination

David P. Sumner
1990 Discrete Mathematics  
This theorem is a direct consequence of Theorem 26 and the observation that the graphs in Fig. 8 are both nonplanar.  ...  Unfortunately it is impossible to provide a finite forbidden characterization of the entire class of domination perfect graphs.  ...  Fig Fig. 1 shows a selection of examples of 3-critical graphs of small order (at most 9 vertices). Fig. 1 . 1 Fig. 1. Fig. 1 . 1 (contd.) Theorem 4 . 4 d2 s 2 for n >> 2.Proof.  ... 
doi:10.1016/0012-365x(90)90347-k fatcat:tzokafw6qrfu3dqjs4wqtj6rwe

Page 8200 of Mathematical Reviews Vol. , Issue 99m [page]

1999 Mathematical Reviews  
Conjecture A: For every k > 1, there is no k- critical graph of even order. Conjecture B: For all k > 1, every k-critical graph of even order has a 1|-factor.  ...  simple graphs of order ((k + 1)(7k — 1))/2 +2, for every odd k > 3, (iii) k-critical multigraphs of order 12k — 20, for every k >4, (iv) k-critical multigraphs of order 7k + 1, for every odd k > 3.  ... 

Computing Small Unit-Distance Graphs with Chromatic Number 5 [article]

Marijn J.H. Heule
2018 arXiv   pre-print
The lack of a 4-coloring of our graphs is due to a clear pattern enforced on some vertices.  ...  We present a new method for reducing the size of graphs with a given property.  ...  Acknowledgements The author thanks Aubrey de Grey, Jasmin Blanchette, Benjamin Kiesl, Victor Marek, and the reviewer for their valuable input on an earlier draft of this paper.  ... 
arXiv:1805.12181v1 fatcat:sqpymfjkdfhsnc5iomriog5zxq

Circular chromatic number and a generalization of the construction of Mycielski

Peter Che Bor Lam, Wensong Lin, Guohua Gu, Zengmin Song
2003 Journal of combinatorial theory. Series B (Print)  
Because m m ðK 3 Þ; for arbitrary m; is a planar graph with connectivity 3 and maximum degree 4; it provides another counterexample to a question asked by Vince (J. Graph Theory 17 (1993) 349).  ...  For any positive even number nðX2Þ and any nonnegative integer m; we show that w c ðm m ðK n ÞÞ ¼ n þ ð1=tÞ; where t ¼ I2m=nm þ 1: This gives a family of arbitrarily large critical graphs G with high connectivity  ...  Acknowledgments The authors acknowledge the valuable comments and suggestions of the referees.  ... 
doi:10.1016/s0095-8956(03)00070-4 fatcat:nnwxi2n2gnetvosaludb5qbxrm

Page 2449 of Mathematical Reviews Vol. , Issue 2000d [page]

2000 Mathematical Reviews  
M. (4-RDNG-E; Reading); Hilton, A. J. W. (4-RDNG; Reading); Hind, H. R. F. (3-WTRL-B; Waterloo, ON) Totally critical even order graphs. (English summary) J. Combin. Theory Ser.  ...  In this paper, the authors mainly characterize the totally critical graphs G of even order with A(G) > +|V(G)|.  ... 

Page 6090 of Mathematical Reviews Vol. , Issue 2001I [page]

2001 Mathematical Reviews  
order graphs G satisfying |V(G)|/2 < A(G) < 3|V(G)|/4—1 it would also follow that if G were totally critical, then G would be regular.  ...  A com- plete list of all totally critical graphs of order at most ten is given, and some open problems are stated.”  ... 

Edge Criticality in Graph Domination

Jan H van Vuuren
2015 Graphs and Combinatorics  
The classes of q-critical graphs of order n are characterised in this paper for all admissible combinations of values of n and q.  ...  The cardinality of a smallest dominating set of G is called the domination number of G and a nonempty graph G is q-critical if q is the smallest number of arbitrary edges of G whose removal from G necessarily  ...  The anonymous reviewers are also thanked for their suggestions which have improved the quality of the paper.  ... 
doi:10.1007/s00373-015-1607-5 fatcat:xngtovjelnafdmtmgnhmetprfu

Chromatic-Index-Critical Graphs of Orders 11 and 12

G Brinkmann, E Steffen
1998 European journal of combinatorics (Print)  
We prove that there is no chromatic-index-critical graph of order 12, and that there are precisely two non-trivial chromatic index critical graphs on 11 vertices.  ...  Together with known results this implies that there are precisely three non-trivial chromaticindex-critical graphs of order 12.  ...  Recently Gr unewald and Ste en 13] constructed k-critical graphs of even order for each k 3.  ... 
doi:10.1006/eujc.1998.0254 fatcat:o7cymzc22rbrhmrodlxxckjjoa

Monte Carlo Simulation of Small-World Models with Weak Long-Range Interactions [article]

X. Zhang, M. A. Novotny
2006 arXiv   pre-print
Large-scale Monte Carlo simulations, together with scaling, are used to obtain the critical behavior of the Hastings long-range model and two corresponding models based on small-world networks.  ...  These models have combined short- and long-range interactions. For all three models, the critical scaling behavior predicted by Hastings [Phys. Rev. Lett. 91, 098701] is verified.  ...  ACKNOWLEDGEMENTS We acknowledge useful discussions with a number of people, particularly Gyorgy Korniss, Per Arne Rikvold, and Zoltan Toroczkai.  ... 
arXiv:cond-mat/0602286v1 fatcat:la2djkwmqjgotke2w3idtgphcu

The spectral radius of edge chromatic critical graphs

Lihua Feng, Jianxiang Cao, Weijun Liu, Shifeng Ding, Henry Liu
2016 Linear Algebra and its Applications  
A connected graph G with maximum degree ∆ and edge chromatic number χ (G) = ∆ + 1 is called ∆-critical if χ (G − e) = ∆ for every edge e of G.  ...  In this paper, we consider two weaker versions of Vizing's conjecture, which concern the spectral radius ρ(G) and the signless Laplacian spectral radius µ(G) of G.  ...  The authors are grateful to the referee for their valuable comments which resulted in a significant improvement of this paper. Feng  ... 
doi:10.1016/j.laa.2015.11.019 fatcat:gh27xwbbi5f6dgvqgvyenwlxmu

Morphology of small snarks [article]

Ján Mazák, Jozef Rajník, Martin Škoviera
2021 arXiv   pre-print
The aim of this paper is to classify all snarks up to order 36 and explain the reasons of their uncolourability.  ...  Our results reveal that most of the analysed snarks are built up from pieces of the Petersen graph and can be naturally distributed into a small number of classes having the same reason for uncolourability  ...  This can be observed already among snarks of small order: there are exactly 55172 critical snarks of order not exceeding 36, but only 846 of them are strictly critical, just slightly over 1.5 percent [  ... 
arXiv:2112.04335v1 fatcat:jggoupeqwjc5dk4cylbeivqzu4

High connectivity keeping sets in graphs and digraphs

W. Mader
2005 Discrete Mathematics  
We survey results and possible results of the following type: given positive integers n, k, d, is there an integer h d (n, k) so that every n-connected, finite graph (digraph) G of order at least h d (  ...  n, k) contains a set W of exactly k vertices so that G − W is (n − d)-connected?  ...  Theorem 3.13 is equivalent to the case k = 3 of this conjecture. Conjecture 3.14 is not proved for any pair (n, k) with n 3 and k 4.  ... 
doi:10.1016/j.disc.2004.07.032 fatcat:lnbgsxcwxfbwtc7alknixofqmm
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