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On a conjecture of Gallai concerning complete subgraphs of k-critical graphs

H.L. Abbott, B. Zhou
1992 Discrete Mathematics  
Zhou, On a conjecture of Gallai concerning complete subgraphs of k-critical graphs, Discrete Mathematics 100 (1992) 223-228.  ...  Gallai asked whether each k-critical graph of order n contains at most n complete subgraphs of order k -1.  ...  Gallai raised the following problem: Is it true that every k-critical graph of order n contains at most it complete subgraphs of order k -l?  ... 
doi:10.1016/0012-365x(92)90642-s fatcat:mllzaduwhrbw3lnqpdbisngqye

Rainbow Generalizations of Ramsey Theory: A Survey

Shinya Fujita, Colton Magnant, Kenta Ozeki
2010 Graphs and Combinatorics  
In this work, we collect Ramsey-type results concerning rainbow and proper edge colorings of graphs. Editions  ...  In slightly more generality, one could ask for any monochromatic subgraph of a k-edge colored complete graph. More formally, we have the following definition.  ...  Proof: By Theorem 75, the Gallai coloring of K n is constructed by substituting Gallai colored complete graphs into vertices of a 2-colored base graph.  ... 
doi:10.1007/s00373-010-0891-3 fatcat:ytvtt2g4hnhglbyx725xof43de

Page 4148 of Mathematical Reviews Vol. , Issue 93h [page]

1993 Mathematical Reviews  
[Zhou, Bing] (3-TREN) On a conjecture of Gallai concerning complete subgraphs of k- critical graphs.  ...  Gallai conjectured that every k-critical graph on n vertices contains at most n cliques of size k —1. The conjecture is trivially true for k < 3. M.  ... 

Rainbow generalizations of Ramsey theory - a dynamic survey

Shinya Fujita, Colton Magnant, Kenta Ozeki
2014 Theory and Applications of Graphs  
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs.  ...  The authors of [68] also found the Gallai Ramsey numbers for all trees of order at most 6. Regarding paths in general, the following represents the best known bounds.  ...  Acknowledgement The authors would like to thank Maria Axenovich, Tao Jiang and the referee for helpful comments and corrections on the initial publication of this work.  ... 
doi:10.20429/tag.2014.000101 fatcat:zawfrrlt4jc4fbtiebsbpprsli

Inflations of anti-cycles and Hadwiger's Conjecture

Anders Sune Pedersen, Michael D. Plummer, Bjarne Toft
2016 Journal of Combinatorics  
Introduction Hadwiger's Conjecture [9] from 1943 suggests a far reaching generalization of the Four Color Theorem, namely that any k-chromatic graph has the complete graph K k as a minor.  ...  Then H is disconnected by Gallai [8] (Satz E 2 .1). and hence H consists of the join of two non-empty critical subgraphs H 1 and H 2 .  ...  The proof, based on Hall's Theorem, is elementary, but not trivial. Our Theorems 2.1 and 2.6 can then be derived as corollaries of this lemma.  ... 
doi:10.4310/joc.2016.v7.n2.a10 fatcat:gxgcb47unrh35jqjvsolufcxsi

Page 46 of Mathematical Reviews Vol. , Issue 80A [page]

1980 Mathematical Reviews  
The richest chapter is the one on complete subgraphs where for instance the result of Hajnal and Szemerédi on partitioning the vertices of a graph of maximal degree A into (4+ 1) almost equicardinal independent  ...  In contrast to the linear growth rate of the maximum number of k-blocks in a p-vertex graph, it is shown that the maximum number of critical k-vertex-connected subgraphs of an ultrablock of connectivity  ... 

On complete subgraphs of color-critical graphs

Xiang-Ying Su
1995 Discrete Mathematics  
Gallai conjectured that every k-critical graph of order n contains at most n complete (kl)-subgraphs.  ...  In their paper, Abbott and Zhou asked the following question: is it true that the number of complete (k -1)-subgraphs of any k-critical graph G of order n > k is at most n -k + 3 (k > 5)?  ...  Acknowledgement The author sincerely thanks one of the referees for providing the present proof of Lemma 2, which is shorter than the original proof.  ... 
doi:10.1016/0012-365x(94)00017-d fatcat:b6njcj6kyvf3zmycqkk25hogbq

Triangle Packings and Transversals of Some $$K_{4}$$K4-Free Graphs

Andrea Munaro
2018 Graphs and Combinatorics  
Acknowledgements The author would like to thank the anonymous referees for valuable comments improving the structure of the paper and simplifying the proof of Theorem 13.  ...  On the other hand, the recognition of k-Gallai graphs remains a major open problem. Let us now translate Tuza's Conjecture in terms of the triangle graph.  ...  We then rely on the following result by Gallai (see [39] for a short proof and an extension): Theorem 14 (Gallai [13] ) If v is any vertex of a connected θ-critical graph G, then G has a minimum-size  ... 
doi:10.1007/s00373-018-1903-y fatcat:lssr3xle7vh35h2soo4ah4kw3u

Page 2290 of Mathematical Reviews Vol. , Issue 87e [page]

1987 Mathematical Reviews  
(ie. a subgraph of a k-tree) for small k.  ...  87e:05051 orem on perfect graphs (1983) and Berge’s conjectured common generalization of the Gallai- Milgram and Gallai- Roy theorems. There are two slight problems with the revisions.  ... 

Page 27 of Mathematical Reviews Vol. 49, Issue 1 [page]

1975 Mathematical Reviews  
It was conjectured by Griinbaum and Rao that if {d,} and {d,—k} are both graphical then there exists a graph realizing {d,} that has a spanning regular subgraph of degree k.  ...  A note on factor-critical graphs. Studia Sci. Math. Hungar. '7 (1972), 279-280.  ... 

Critically paintable, choosable or colorable graphs

Ayesha Riasat, Uwe Schauz
2012 Discrete Mathematics  
Using a strong version of Brooks' Theorem, we generalize Gallai's Theorem about the structure of the low-degree subgraph of critically k-colorable graphs, and introduce a more adequate lowest-degree subgraph  ...  We also show that on a fixed given surface, there are only finitely many critically k-paintable/k-choosable/ k-colorable graphs, if k ≥ 6.  ...  Acknowledgments We want to thank the Abdus Salam School of Mathematical Sciences for providing a good research environment.  ... 
doi:10.1016/j.disc.2012.07.035 fatcat:esfrt5d2qvarja6h7wobmklnai

Page 1323 of Mathematical Reviews Vol. , Issue 96c [page]

1996 Mathematical Reviews  
Gallai conjectured that every k-critical graph of order n contains at most n complete (k — 1)-subgraphs.  ...  In their paper, Abbott and Zhou asked the following question: Is it true that the number of complete (k — 1)-subgraphs of any k-critical graph G of order n >k is at most n —k +3 (k > 5)?  ... 

Page 2454 of Mathematical Reviews Vol. , Issue 94e [page]

1994 Mathematical Reviews  
A graph G is k-critical if x(G) = k and for every proper subgraph H of G, x(H) < k, where x(G) is the chromatic number of G.  ...  The author shows, by means of an example, that a conjecture concerning the Tutte polynomial of a graph—namely that the coefficients form two-way unimodal sequences—is false. R. C.  ... 

On 4-critical planar graphs with high edge density

Gerhard Koester
1991 Discrete Mathematics  
Four problems of Grtinbaum concerning 4-critical planar graphs with only vertices of valence higher than three are treated  ...  ., On 4-critical planar graphs with high edge density, Discrete Mathematics 98 (1991) 147-151.  ...  An old conjecture of Gallai [2] that every 4-critical planar graph G contains e 5 2v -2 edges (that means d(G) d L d S c 2) was recently disproved by the present author [4] giving a 4-critical planar  ... 
doi:10.1016/0012-365x(91)90039-5 fatcat:wolrqgixfjc5hkhd6oawnabqfm

On color critical graphs

Vojtech Rödl, Zsolt Tuza
1985 Journal of combinatorial theory. Series B (Print)  
Denote by f(l, k, n) the largest positive integer such that any k-critical graph with an vertices contains an Z-critical subgraph with af(Z, k, n) vertices. Conjecture.  ...  The statement concerning the graphs Gk,, can be obtained easily from the Turin theorem. 1 THEOREM 1.4. If x(G) = k + 1, x(G') = 1, G' is a subgraph of G, then II?(G) -E(G')I > t(k+ 1, I+ 1).  ... 
doi:10.1016/0095-8956(85)90066-8 fatcat:opuk5us2gnhtfe7xa6czinsoyq
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