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A new lower bound on the number of edges in colour-critical graphs and hypergraphs

2003
*
Journal of combinatorial theory. Series B (Print)
*

*A*

*graph*G is called k-

*critical*if it has chromatic

*number*k; but every proper subgraph

*of*G is ðk

*À*1Þ-

*colourable*. ... We prove that every k-

*critical*

*graph*ðkX6Þ

*on*nXk þ 2 vertices has at least 1 2 ðk

*À*1 þ kÀ3 ðkÀcÞðkÀ1ÞþkÀ3 Þn

*edges*where c ¼ ðk

*À*5Þð 1 2

*À*1 ðkÀ1ÞðkÀ2Þ Þ: This improves earlier

*bounds*established by ...

*Critical*

*graphs*were first defined

*and*used by Dirac [5]

*in*1951.

*In*

*the*present paper

*a*

*new*

*lower*

*bound*for

*the*

*number*

*of*

*edges*

*in*

*a*k-

*critical*

*graph*

*on*n vertices is established. ...

##
###
On the Number of Edges in Hypergraphs Critical with Respect to Strong Colourings

2000
*
European journal of combinatorics (Print)
*

*In*

*the*case when

*the*generated

*graph*

*of*

*the*

*hypergraph*has

*bounded*clique

*number*, we give

*a*

*lower*

*bound*that is valid for sufficiently large k

*and*is asymptotically tight

*in*k; this

*bound*also holds for ... We estimate

*the*minimum

*number*

*of*

*edges*possible

*in*

*a*k-

*critical*t-uniform

*hypergraph*with

*a*given

*number*

*of*vertices. ...

*The*work

*of*this author was also partly supported by grants 96-01-01614

*and*97-01-01075

*of*

*the*Russian Foundation for Fundamental Research. ...

##
###
Page 2606 of Mathematical Reviews Vol. , Issue 96e
[page]

1996
*
Mathematical Reviews
*

Summary: “

*A*Hoang*graph*G = (V,E) is*a**graph*whose*edges*may be*coloured*with two*colours*(red*and*white) such that no induced path*on*four vertices has wings*of**the*same*colour*. ...*The*author introduces*the*notion*of**a*co-*edge**in**a**hypergraph**and*defines what is meant by*a*free coloring.*The*co-*edges**in**a*free coloring contain at least two vertices*of**the*same color. ...##
###
Connected τ -critical hypergraphs of minimal size

2005
*
Discrete Mathematics & Theoretical Computer Science
*

International audience

doi:10.46298/dmtcs.3397
fatcat:xzevnh6uvzdb5jkasffq24jfqa
*A**hypergraph*$\mathscr{H}$ is $τ$ -*critical*if $τ (\mathscr{H}-E) < τ (\mathscr{H})$ for every*edge*$E ∈\mathscr{H}$, where $τ (\mathscr{H})$ denotes*the*transversal*number**of*$\ ... It can be shown that*a*connected $τ$ -*critical**hypergraph*$\mathscr{H}$ has at least $2τ (\mathscr{H})-1$*edges*; this generalises*a*classical theorem*of*Gallai*on*$χ$ -vertex-*critical**graphs*with connected ... We first present*a*sharp*lower**bound**on**the**number**of**edges**in**a*connected τ -*critical**hypergraph*,*and*then investigate*the*cases where equality is attained. ...##
###
Page 2334 of Mathematical Reviews Vol. , Issue 82f
[page]

1982
*
Mathematical Reviews
*

He also establishes an analogous result for

*hypergraphs**and*gives best possible*lower**bounds**in*certain special cases. 82f:05030 J. W. Moon (Edmonton, Alta.) ... Given*a*cellularly imbedded*graph*L—S*and*an assignment*of*voltages to*the**edges**of*L*in**a*group,*one*may construct*a*branched covering p:S—S*of*surfaces, such that*the**graph*p™'(L) is*a*covering*graph*...##
###
Path Coupling Using Stopping Times
[chapter]

2005
*
Lecture Notes in Computer Science
*

We show that

doi:10.1007/11537311_3
fatcat:nhgjsh2c3nftllhewoagseffru
*the*Glauber dynamics for independent sets*in**a**hypergraph*mixes rapidly as long as*the*maximum degree ∆*of**a*vertex*and**the*minimum size m*of*an*edge*satisfy m ≥ 2∆ + 1. ... We also state results that*the*Glauber dynamics for proper q-*colourings**of**a**hypergraph*mixes rapidly if m ≥ 4*and*q > ∆,*and*if m = 3*and*q ≥ 1.65∆. ... Acknowledgments We are grateful to Tom Hayes for commenting*on*an earlier draft*of*this paper,*and*to Mary Cryan for useful discussions at an early stage*of*this work. ...##
###
Page 6546 of Mathematical Reviews Vol. , Issue 2003i
[page]

2003
*
Mathematical Reviews
*

*The*main feature is that mixed

*hypergraphs*represent structures

*in*which problems

*on*both

*the*minimum

*and*maximum

*number*

*of*

*colours*occur. ... It belongs

*on*

*the*shelves

*of*everyone who works not only

*in*

*graph*

*and*

*hypergraph*theory, but more generally

*in*discrete mathematics. ...

##
###
Page 2347 of Mathematical Reviews Vol. , Issue 2003d
[page]

2003
*
Mathematical Reviews
*

*A*

*hypergraph*H is said to be

*a*

*hypergraph*with p-definable

*edges*if any

*edge*

*of*H contains at least p+ 1 vertices

*and*any p vertices

*of*H are contained

*in*not more than

*one*

*edge*. ...

*A*

*graph*is 3-domination-

*critical*if its domination

*number*is 3

*and*

*the*addition

*of*any

*edge*reduces

*the*domination

*number*to 2. ...

##
###
On the minimum degree of minimal Ramsey graphs for multiple colours
[article]

2015
*
arXiv
*
pre-print

*A*

*graph*G is r-Ramsey for

*a*

*graph*H, denoted by G→ (H)_r, if every r-

*colouring*

*of*

*the*

*edges*

*of*G contains

*a*monochromatic copy

*of*H. ... We also give an upper

*bound*

*on*s_r(K_k) which is polynomial

*in*both r

*and*k,

*and*we determine s_r(K_3) up to

*a*factor

*of*log r. ... Therefore,

*the*

*lower*

*bound*

*on*s r (K 3 ) does not necessarily imply

*a*similar

*lower*

*bound*

*on*s r (K k ). We can

*in*fact only prove

*a*super-quadratic

*lower*

*bound*

*on*s r (K k ) that is slightly weaker. ...

##
###
On the minimum degree of minimal Ramsey graphs for multiple colours

2016
*
Journal of combinatorial theory. Series B (Print)
*

*A*

*graph*G is r-Ramsey for

*a*

*graph*H, denoted by G → (H) r , if every r-

*colouring*

*of*

*the*

*edges*

*of*G contains

*a*monochromatic copy

*of*H. ... We also give an upper

*bound*

*on*s r (K k ) which is polynomial

*in*both r

*and*k,

*and*we determine s r (K 3 ) up to

*a*factor

*of*log r. ... Therefore,

*the*

*lower*

*bound*

*on*s r (K 3 ) does not necessarily imply

*a*similar

*lower*

*bound*

*on*s r (K k ). We can

*in*fact only prove

*a*super-quadratic

*lower*

*bound*

*on*s r (K k ) that is slightly weaker. ...

##
###
Rainbow Turán Problems

2006
*
Combinatorics, probability & computing
*

For

doi:10.1017/s0963548306007760
fatcat:wap4bn3jt5dw3mq4tvjqem7kdy
*a*fixed*graph*H, we define*the*rainbow Turán*number*ex * (n, H) to be*the*maximum*number**of**edges**in**a**graph**on*n vertices that has*a*proper*edge*-*colouring*with no rainbow H. ... Recall that*the*(ordinary) Turán*number*ex(n, H) is*the*maximum*number**of**edges**in**a**graph**on*n vertices that does not contain*a*copy*of*H. ... Thus*a**bound**on**the**number**edges**in**a**graph*with no rainbow C 2k gives*a**bound**on**the*size*of**a*B * k -set.*A*related*and*more commonly studied condition is*the*following. ...##
###
Covering graphs by monochromatic trees and Helly-type results for hypergraphs
[article]

2020
*
arXiv
*
pre-print

How many monochromatic paths, cycles or general trees does

arXiv:1902.05055v4
fatcat:ljact63gufg23cbddedlwnki4q
*one*need to cover all vertices*of**a*given r-*edge*-*coloured**graph*G? ... Roughly speaking, this question asks for*the*maximum*number**of*vertices needed to cover all*the**edges**of**a**hypergraph*H if it is known that any collection*of**a*few*edges**of*H has*a*small cover. ...*In*particular, we are grateful for*a*suggestion*on*how to rewrite*the*proof*of*Lemma 4.3 to make it easier to follow. ...##
###
Path Coupling Using Stopping Times and Counting Independent Sets and Colourings in Hypergraphs
[article]

2005
*
arXiv
*
pre-print

We show that

arXiv:math/0501081v2
fatcat:4sjn6ofyvbdbjf5cphysail6k4
*the*Glauber dynamics for independent sets*in**a**hypergraph*mixes rapidly as long as*the*maximum degree Delta*of**a*vertex*and**the*minimum size m*of*an*edge*satisfy m>= 2Delta+1. ... We also show that*the*Glauber dynamics for proper q-*colourings**of**a**hypergraph*mixes rapidly if m>= 4*and*q > Delta,*and*if m=3*and*q>=1.65Delta. ... Acknowledgments We are grateful to Tom Hayes for commenting*on*an earlier draft*of*this paper,*and*to Mary Cryan for useful discussions at an early stage*of*this work. ...##
###
The colour theorems of Brooks and Gallai extended

1996
*
Discrete Mathematics
*

*One*

*of*

*the*basic results

*in*

*graph*

*colouring*is Brooks' theorem [-4] which asserts that

*the*chromatic

*number*

*of*every connected

*graph*, that is not

*a*complete

*graph*or an odd cycle, does not exceed its maximum ...

*In*this note, we use

*a*reduction idea

*in*order to give

*a*

*new*short proof

*of*this result

*and*to extend it to

*hypergraphs*. ... For

*colour*-

*critical*

*graphs*

*the*above Theorem was proved by Gallai [6] . Moreover, using his result, Gallai established

*a*

*lower*

*bound*for

*the*

*number*

*of*

*edges*

*of*

*a*k-

*colour*-

*critical*

*graph*. ...

##
###
Page 4111 of Mathematical Reviews Vol. , Issue 98G
[page]

1998
*
Mathematical Reviews
*

(SGP-SING; Singapore)

*On**graphs**in*which any pair*of**colour*classes but*one*induces*a*tree. ... They then establish*bounds**on**the*size*and*maximum degree*of**a*connected*graph**in*terms*of*its order*and*diameter*and*show that these same*bounds*are sharp for L-*graphs*as well. ...
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