A copy of this work was available on the public web and has been preserved in the Wayback Machine. The capture dates from 2019; you can also visit the original URL.
The file type is `application/pdf`

.

## Filters

##
###
Gallai's problem on Dirac's construction

1992
*
Discrete Mathematics
*

.,

doi:10.1016/0012-365x(92)90615-m
fatcat:ptkyvx3n5jdjtewgqcl2kgxzee
*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. ...##
###
Crossings, Colorings, and Cliques

2009
*
Electronic Journal of Combinatorics
*

lower bounds by Pach et al.

doi:10.37236/134
fatcat:5fhmswdmwjdytm4lo7lw7xcxve
*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. ...##
###
Crossings, colorings, and cliques
[article]

2010
*
arXiv
*
pre-print

by Pach et.al.

arXiv:1006.3783v1
fatcat:7ofzuf66zvf6zca3dt675zxate
*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. ...##
###
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

doi:10.1002/9781118032497.ch10
fatcat:374tktuvgvekni4fnz3dgbytjm
*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 ...##
###
Critically paintable, choosable or colorable graphs

2012
*
Discrete Mathematics
*

Using a strong version of Brooks' Theorem, we generalize

doi:10.1016/j.disc.2012.07.035
fatcat:esfrt5d2qvarja6h7wobmklnai
*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 ...##
###
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

2017
*
Combinatorica
*

essentially solved the

doi:10.1007/s00493-016-3584-6
fatcat:v5ihcsy7abferjuqmeoifd4jxu
*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. ...##
###
Edge Lower Bounds for List Critical Graphs, via Discharging
[article]

2016
*
arXiv
*
pre-print

essentially solved the

arXiv:1602.02589v1
fatcat:mwp3y5rdhjfv7ovzlhgoxpti3e
*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.) ...##
###
Edge colorings of complete graphs without tricolored triangles

2004
*
Journal of Graph Theory
*

This

doi:10.1002/jgt.20001
fatcat:hj6jkrcrnrh6foqfk54llol5ma
*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. ...##
###
Towards The Albertson Conjecture
[article]

2009
*
arXiv
*
pre-print

It is clear that our improvement

arXiv:0909.0413v1
fatcat:x733wowu75datajdln2fy2532i
*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. ...##
###
Coloring Vertices and Faces of Locally Planar Graphs

2006
*
Graphs and Combinatorics
*

A graph G drawn

doi:10.1007/s00373-006-0653-4
fatcat:irjfjlxi5vgtpfcp75mgl653jq
*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. ...##
###
Color-Critical Graphs Have Logarithmic Circumference
[article]

2011
*
arXiv
*
pre-print

We thus settle the

arXiv:0908.3169v2
fatcat:taqx3qtehbg4dbs2oxa36m7slu
*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] . ...##
###
Color-critical graphs have logarithmic circumference

2011
*
Advances in Mathematics
*

We thus settle the

doi:10.1016/j.aim.2011.05.001
fatcat:vsc22c7uhzfv7hvpb7goiob2b4
*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] . ...##
###
Color-Critical Graphs and Hypergraphs with Few Edges: A Survey
[chapter]

2006
*
Bolyai Society Mathematical Studies
*

The current situation with bounds

doi:10.1007/978-3-540-32439-3_9
fatcat:6hmpcen2c5godpbueoyqnigzye
*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. ...##
###
On the edge-density of 4-critical graphs

2009
*
Combinatorica
*

Gallai conjectured that every 4-critical graph

doi:10.1007/s00493-009-2267-y
fatcat:u4khproaprbr3ggtki44eflsx4
*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. ...
« Previous

*Showing results 1 — 15 out of 23 results*