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

Paul Erdős, Ralph Faudree, Talmage James Reid, Richard Schelp, William Staton
1995 Discrete Mathematics  
We investigate whether K,-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.  ... 
doi:10.1016/0012-365x(93)e0226-t fatcat:kwabh5dx4jfzlpeip7e7ng24su

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.  ... 

The maximum number of triangles in a K4-free graph

Jürgen Eckhoff
1999 Discrete Mathematics  
Let v, e 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  ... 
doi:10.1016/s0012-365x(98)00120-4 fatcat:l3c4cu3bpjdanh2hv5nv7rzc5y

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

Pavel Hrubeš
2007 Journal of Symbolic Logic (JSL)  
We give an exponential 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  ... 
doi:10.2178/jsl/1191333849 fatcat:5osv2td5ovgrzoijjzmaasrxja

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.  ... 

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

Michael O. Albertson, Ruth Haas
1996 Discrete Mathematics  
This paper investigates 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.  ... 
doi:10.1016/0012-365x(94)00254-g fatcat:ktiirpqqc5frpkp67cbburvpaq

Graphs without large triangle free subgraphs

B. Bollobás, H.R. Hind
1991 Discrete Mathematics  
Bollobas, B. 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.  ... 
doi:10.1016/0012-365x(91)90042-z fatcat:ca2rl4lalferlio4r3dm3kj27q

On the Number of Edge-Disjoint Triangles in K4-Free Graphs

Ervin Győri, Balázs Keszegh
2016 Combinatorica  
We show the quarter of a century old conjecture that every K 4 -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.  ... 
doi:10.1007/s00493-016-3500-0 fatcat:ohygtvbzsbflbhyazcqzecy32q

On applications of Razborov's flag algebra calculus to extremal 3-graph theory [article]

Victor Falgas-Ravry, Emil R. Vaughan
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.  ... 
arXiv:1110.1623v2 fatcat:egysyroqsrbxfp5ju62wtdhsem

Turán H-densities for 3-graphs [article]

Victor Falgas-Ravry, Emil R. Vaughan
2012 arXiv   pre-print
The 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.  ... 
arXiv:1201.4326v1 fatcat:id5mlvwvynal3gd6e73v5n2rve

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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