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Degree sequence and independence in K(4)-free graphs

1995
*
Discrete Mathematics
*

We investigate whether K,-

doi:10.1016/0012-365x(93)e0226-t
fatcat:kwabh5dx4jfzlpeip7e7ng24su
*free**graphs*with few repetitions*in*the degree sequence may have*independence*number o(n). ... We settle the cases r = 3*and*r >/5,*and*give partial results for the very interesting case r=4. ... D Now*in*the case of*K4*-*free**graphs*withf(G)= 3 we are unable to exhibit a precise*lower**bound*for fl, but using a result of Szemer6di we can show that fl=o(n) is not possible. Theorem 7. ...##
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Page 4775 of Mathematical Reviews Vol. , Issue 98H
[page]

1998
*
Mathematical Reviews
*

) Finding

*independent*sets*in**K4*-*free*4-regular connected*graphs*. ... They construct a family of*graphs**in*%, with*independence*number n, thereby showing that the*independence*ratio a/n of*graphs**in*¥ has*lower**bound*lying between +*and*3. ...##
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The maximum number of triangles in a K4-free graph

1999
*
Discrete Mathematics
*

Let v, e

doi:10.1016/s0012-365x(98)00120-4
fatcat:l3c4cu3bpjdanh2hv5nv7rzc5y
*and*t denote the number of vertices, edges*and*triangles, respectively, of a*K4*-*free**graph*. Fisher (1988) proved that t<,(e/3) 3/2,*independently*of v. ... We also show that certain 'gaps' occur*in*the set of triples (v, e, t) realizable by*K4*-*free**graphs*. (~) 1999 Elsevier Science B.V. ... Acknowledgements We are indebted to Klaus-Peter Nischke for providing us, among other data, with a computer list of all realizable triples of Ka-*free**graphs**on*up to 10 vertices, to Gerrit van Ophuysen ...##
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Page 8088 of Mathematical Reviews Vol. , Issue 2003k
[page]

2003
*
Mathematical Reviews
*

*In*this paper the authors derive

*bounds*

*on*the

*sizes*of tactical configurations of large girth. This is equivalent to

*bounding*the number of edges

*in*biregular bipartite

*graphs*of large girth. ... /7, where c is a positive constant

*independent*of v, ¢ = a(3g —8)/4 for g =0 (mod 4),

*and*¢ = a(3g — 10)/4 for g=2 (mod 4). For some values of g

*and*a this general

*lower*

*bound*is refined. ...

##
###
Page 7629 of Mathematical Reviews Vol. , Issue 2000k
[page]

2000
*
Mathematical Reviews
*

We prove that a generalization of the residue, the k- residue of a

*graph*, yields a*lower**bound**on*the k-*independence*number. The new*bound*strengthens a*bound*of Y. Caro*and*Z. Tuza [J. ...*Graph*Theory 15 (1991), no. 1, 39-64; MR 92d:05073] proved that the residue of a simple*graph*G is a*lower**bound**on*its*independence*number a(G). ...##
###
Lower bounds for modal logics

2007
*
Journal of Symbolic Logic (JSL)
*

We give an exponential

doi:10.2178/jsl/1191333849
fatcat:5osv2td5ovgrzoijjzmaasrxja
*lower**bound**on*number of proof-lines*in*the proof system K of modal logic, i.e., we give an example of K-tautologies ψ1, ψ2, ... s.t. every K-proof of ψ i must have a number of proof-lines ... exponential*in*terms of the*size*of ψ i . ... The same*lower**bound*is valid also*in*the case of*K4*, Gödel-Löb's logic, S*and*S4, by showing that the additional axioms do not lead to shortening of proofs*on*tautologies of modal depth*one*(as far as ...##
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Page 818 of Mathematical Reviews Vol. , Issue 2002B
[page]

2002
*
Mathematical Reviews
*

*In*this paper we establish the best possible upper

*bounds*

*on*the ratios y(G)/ra(G)

*and*»(G)/ir(G)

*in*the class of @F¥@-

*graphs*. ... *O< \

*On*| While

*lower*

*bounds*for the density of a 1-identifying code are known for the other grids,

*in*this paper the authors obtain the

*bound*s for the density for the hexagonal grid. ...

##
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Page 1296 of Mathematical Reviews Vol. , Issue 91C
[page]

1991
*
Mathematical Reviews
*

It is shown that the smallest root of the dependence polynomial is real

*and*positive, which results*in*implementing a new*lower**bound*for the number of triangles*in*a*K4*-*free**graph*. ... T. (1-AZS) The maximum number of edges*in*2K>-*free**graphs*of*bounded*degree. Discrete Math. 81 (1990), no. 2, 129-135. ...##
###
Parsimonious edge coloring

1996
*
Discrete Mathematics
*

This paper investigates

doi:10.1016/0012-365x(94)00254-g
fatcat:ktiirpqqc5frpkp67cbburvpaq
*lower**bounds*for 7 together with infinite families of 13*graphs**in*which y is*bounded*away from 1. ... For instance, if G is cubic, then 7 >i ]7;*and*there _<25 exists an infinite family of 3-connected cubic*graphs**in*which 7 ~ 27. ... Acknowledgements We are grateful to Stephen Locke for providing information about prior work*on*this topic. ...##
###
Graphs without large triangle free subgraphs

1991
*
Discrete Mathematics
*

Bollobas, B.

doi:10.1016/0012-365x(91)90042-z
fatcat:ca2rl4lalferlio4r3dm3kj27q
*and*H.R. Hind,*Graphs*without large triangle*free*subgraphs, Discrete Mathematics 87 (1991) 119-131. ... The main aim of the paper is to show that for 2 < r <s*and*large enough n, there are*graphs*of order n*and*clique number less than s*in*which every set of vertices, which is not too small, spans a clique ... For H* E I@(n, p)*, define the*graph*GH.*on*the vertex set V which has an edge joining vertex i to vertex j if {i, j} is contained*in*a hyperedge of H*. Clearly such a*graph*G,. is*K4*-*free*. ...##
###
On the Number of Edge-Disjoint Triangles in K4-Free Graphs

2016
*
Combinatorica
*

We show the quarter of a century old conjecture that every K 4 -

doi:10.1007/s00493-016-3500-0
fatcat:ohygtvbzsbflbhyazcqzecy32q
*free**graph*with n vertices*and*⌊n 2 /4⌋ + k edges contains k pairwise edge disjoint triangles. ... Theorem 1.7.) for a (not optimal)*lower**bound**on*t by a function of e, n. ... is an*independent*set*and*the union of the*size*1*and**size*2 cliques of P is triangle-*free*. ...##
###
On applications of Razborov's flag algebra calculus to extremal 3-graph theory
[article]

2012
*
arXiv
*
pre-print

*In*this paper, we prove several new Turán density results for 3-

*graphs*with

*independent*neighbourhoods. ... We give several new constructions, conjectures

*and*

*bounds*for Turán densities of 3-

*graphs*which should be of interest to researchers

*in*the area. ... Acknowledgements The authors would like to thank Peter Keevash

*and*Mark Walters for their helpful comments

*and*discussions. ...

##
###
Turán H-densities for 3-graphs
[article]

2012
*
arXiv
*
pre-print

The

arXiv:1201.4326v1
fatcat:id5mlvwvynal3gd6e73v5n2rve
*lower**bound**in*this case comes from a random geometric construction strikingly different from previous known extremal examples*in*3-*graph*theory. ... Given an $r$-*graph*$H$*on*$h$ vertices,*and*a family $\mathcal{F}$ of forbidden subgraphs, we define $\ex_{H}(n, \mathcal{F})$ to be the maximum number of induced copies of $H$*in*an $\mathcal{F}$-*free*... Acknowledgement We would like to thank Colin Reid for coming up with the idea that led to the*lower**bound*construction*in*Theorem 18. ...##
###
Page 3601 of Mathematical Reviews Vol. , Issue 2001F
[page]

2001
*
Mathematical Reviews
*

We also give some

*lower**bounds**on*the number of edges of gossip*graphs*which improve the*ones*given by [G. Fertin, op. cit.]. ... We also show that, when applicable, this method gives the best-known upper*bounds**on*the*size*of minimum gossip*graphs**in*most cases, either improving or match- ing them. ...##
###
Page 2091 of Mathematical Reviews Vol. , Issue 98D
[page]

1998
*
Mathematical Reviews
*

Summary: “

*In*this paper we show the number of 4-cliques*in*a 4-partite*graph*G of*size*e is*bounded*above by e? ... Summary: “We give an exponential*lower**bound*for the maximum number of chords*in*a cycle of a*graph*G*in*terms of the minimum degree of G*and*the girth of G. ...
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