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Gallai's problem on Dirac's construction
1992
Discrete Mathematics
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., 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
2009
Electronic Journal of Combinatorics
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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]
2010
arXiv
pre-print
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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
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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
2012
Discrete Mathematics
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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
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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
2017
Combinatorica
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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]
2016
arXiv
pre-print
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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
2004
Journal of Graph Theory
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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]
2009
arXiv
pre-print
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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
2006
Graphs and Combinatorics
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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]
2011
arXiv
pre-print
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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
2011
Advances in Mathematics
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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]
2006
Bolyai Society Mathematical Studies
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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
2009
Combinatorica
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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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