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Gallai's problem on Dirac's construction

D.A. Youngs
1992 Discrete Mathematics  
., Gallai's problem on Dirac's construction, Discrete Mathematics 101 (1992) 343-350.  ...  Acknowledgement The author wishes to thank Bjarne Toft for some enlightening conversations concerning this problem.  ...  The following problem, thought to have been posed by T. Gallai in 1969 , and based on a construction of G.A.  ... 
doi:10.1016/0012-365x(92)90615-m fatcat:ptkyvx3n5jdjtewgqcl2kgxzee

Crossings, Colorings, and Cliques

Michael O. Albertson, Daniel W. Cranston, Jacob Fox
2009 Electronic Journal of Combinatorics  
lower bounds by Pach et al. on the crossing number of graphs in terms of the number of edges and vertices.  ...  In this paper, we prove the conjecture for $7 \leq r \leq 12$ using results of Dirac; Gallai; and Kostochka and Stiebitz that give lower bounds on the number of edges in critical graphs, together with  ...  We thank Sasha Kostochka for helpful discussions on excess in critical graphs.  ... 
doi:10.37236/134 fatcat:5fhmswdmwjdytm4lo7lw7xcxve

Crossings, colorings, and cliques [article]

Michael O. Albertson, Daniel W. Cranston, Jacob Fox
2010 arXiv   pre-print
by Pach et.al. on the crossing number of graphs in terms of the number of edges and vertices.  ...  In this paper, we prove the conjecture for 7 ≤ r ≤ 12 using results of Dirac; Gallai; and Kostochka and Stiebitz that give lower bounds on the number of edges in critical graphs, together with lower bounds  ...  We thank Sasha Kostochka for helpful discussions on excess in critical graphs.  ... 
arXiv:1006.3783v1 fatcat:7ofzuf66zvf6zca3dt675zxate

Algorithms [chapter]

2011 Graph Coloring Problems  
11.1 Direct Product 180 11.2 Wreath Product 7 181 11.3 A Very Strong Product 182 11.4 Gallai's Problem on Dirac's Construction 183 11.5 Hajos Versus Ore 183 11.6 Length of Hajos Proofs  ...  184 11.7 Hajos Constructions of Critical Graphs 185 11.8 Construction of Hajos Generalized by Dirac 185 11.9 Four-Chromaticity in Terms of 3-Colorability 186 Bibliography 187 12 Edge Colorings  ... 
doi:10.1002/9781118032497.ch10 fatcat:374tktuvgvekni4fnz3dgbytjm

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 extend results about critically k-colorable graphs to choosability and paintability (list colorability and on-line list colorability).  ...  Even Dirac's very basic construction of (k 1 + k 2 )-critical graphs G out of k 1 -critical and k 2 -critical graphs G 1 and G 2 , by just taking the complete join, does not work for choosability and paintability  ... 
doi:10.1016/j.disc.2012.07.035 fatcat:esfrt5d2qvarja6h7wobmklnai

Page 1780 of Mathematical Reviews Vol. , Issue 93d [page]

1993 Mathematical Reviews  
Gallai’s problem on Dirac’s construction. Special volume to mark the centennial of Julius Petersen’s “Die Theorie der regularen Graphs”, Part II. Discrete Math. 101 (1992), no. 1-3, 343-350.  ...  The author considers the following problem, which is thought to have been proposed by T. Gallai, based on a construction by G. A. Dirac: Suppose that the graph K is a join of G and H.  ... 

Edge lower bounds for list critical graphs, via discharging

Daniel W. Cranston, Landon Rabern
2017 Combinatorica  
essentially solved the problem.  ...  In this paper, we improve the best lower bound on the number of edges in a k-list-critical graph.  ...  this problem follow immediately from this theorem.  ... 
doi:10.1007/s00493-016-3584-6 fatcat:v5ihcsy7abferjuqmeoifd4jxu

Edge Lower Bounds for List Critical Graphs, via Discharging [article]

Daniel W. Cranston, Landon Rabern
2016 arXiv   pre-print
essentially solved the problem.  ...  In this paper, we improve the best lower bound on the number of edges in a k-list-critical graph.  ...  ., one counted by q(T ). (For comparison with Gallai's bound, we will have ǫ < k−1 k 2 −3 < γ.)  ... 
arXiv:1602.02589v1 fatcat:mwp3y5rdhjfv7ovzlhgoxpti3e

Edge colorings of complete graphs without tricolored triangles

András Gyárfás, Gábor Simonyi
2004 Journal of Graph Theory  
This construction is best possible as shown by the next theorem (and provides a negative answer to Problem 3.3b in Ref. [2] ). Theorem 3.1.  ...  In this paper we look at some Ramsey-type problems for Gallai colorings. The first problems of this type were studied by Erdó´s, Simonovits and Sós in Ref.  ... 
doi:10.1002/jgt.20001 fatcat:hj6jkrcrnrh6foqfk54llol5ma

Towards The Albertson Conjecture [article]

János Barát, Géza Tóth
2009 arXiv   pre-print
It is clear that our improvement on Gallai's result relies on the fact that Kostochka and Stiebitz improved Dirac's result.  ...  Efforts to solve the Four Color Problem had a great effect on the development of graph theory, and it is one of the most important theorems of the field.  ... 
arXiv:0909.0413v1 fatcat:x733wowu75datajdln2fy2532i

Coloring Vertices and Faces of Locally Planar Graphs

Michael O. Albertson, Bojan Mohar
2006 Graphs and Combinatorics  
A graph G drawn on a surface S is said to be 1-embedded in S if every edge crosses at most one other edge. Borodin proved that if G is 1-embedded in the plane, then χ(G) ≤ 6.  ...  Open Questions We summarize some open problems related to vertex-face colorings of embedded graphs. Question 1. If G is planar, is ch vf (G) ≤ 6 (or 7)?  ...  If G is embedded in S, then the natural construction of superimposing the dual of G onto the embedding of G and adding the vertexface incidences gives a 1-embedding of G vf in S.  ... 
doi:10.1007/s00373-006-0653-4 fatcat:irjfjlxi5vgtpfcp75mgl653jq

Color-Critical Graphs Have Logarithmic Circumference [article]

Asaf Shapira, Robin Thomas
2011 arXiv   pre-print
We thus settle the problem of bounding the minimal circumference of k-critical graphs, raised by Dirac in 1952 and Kelly and Kelly in 1954.  ...  We prove that every k-critical graph on n vertices has a cycle of length at least log n/(100log k), improving a bound of Alon, Krivelevich and Seymour from 2000.  ...  We are grateful to Michael Krivelevich for answering many questions related to this paper and especially for sharing his English translation of the construction of Gallai from [7] .  ... 
arXiv:0908.3169v2 fatcat:taqx3qtehbg4dbs2oxa36m7slu

Color-critical graphs have logarithmic circumference

Asaf Shapira, Robin Thomas
2011 Advances in Mathematics  
We thus settle the problem of bounding the minimal circumference of k-critical graphs, raised by Dirac in 1952 and Kelly and Kelly in 1954.  ...  We prove that every k-critical graph on n vertices has a cycle of length at least log n/(100 log k), improving a bound of Alon, Krivelevich and Seymour from 2000.  ...  Acknowledgment We are grateful to Michael Krivelevich for answering many questions related to this paper and especially for sharing his English translation of the construction of Gallai from [8] .  ... 
doi:10.1016/j.aim.2011.05.001 fatcat:vsc22c7uhzfv7hvpb7goiob2b4

Color-Critical Graphs and Hypergraphs with Few Edges: A Survey [chapter]

Alexandr Kostochka
2006 Bolyai Society Mathematical Studies  
The current situation with bounds on the smallest number of edges in colorcritical graphs and hypergraphs is discussed.  ...  Coloring deals with the fundamental problem of partitioning a set of objects into classes that avoid certain conflicts. Many timetabling, sequencing, and scheduling problems are of this nature.  ...  For k ≥ 6 and arbitrary g, this implies that the problem of testing k-colorability is solvable in polynomial time for graphs that embed on the orientable surface of genus g.  ... 
doi:10.1007/978-3-540-32439-3_9 fatcat:6hmpcen2c5godpbueoyqnigzye

On the edge-density of 4-critical graphs

Babak Farzad, Michael Molloy
2009 Combinatorica  
Gallai conjectured that every 4-critical graph on n vertices has at least 5 3 n − 2 3 edges.  ...  Acknowledgement We are grateful to Michael Steibitz for several helpful comments on this proof.  ...  For example, most known lower bounds on the number of edges in k-critical graphs are based on Gallai's characterization of the low-vertex subgraph.  ... 
doi:10.1007/s00493-009-2267-y fatcat:u4khproaprbr3ggtki44eflsx4
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