Cycles of even length in graphs

Bondy, J. A.; Simonovits, Miklós

doi:10.1016/0095-8956(74)90052-5

Cited by 362 publications

(334 citation statements)

References 6 publications

Supporting

Mentioning

326

Contrasting

Order By: Relevance

“…, k, and hence JB,I = r, r, • • -rk > a, contradicting our assumption that H contains no set of it independent vertices . I We note that for the cure where in is even, an improvement of (1 .1) is already available from a result of Bondy and Simonovits [4], used in conjunction with Turán's theorem . With m = 21, the upper bound obtained this way is, asymptotically, (200 ln)' '-'~'(I fixed, it -co) .…”

Section: Notationmentioning

confidence: 90%

On cycle—Complete graph ramsey numbers

Erdös

Faudree

Rousseau

et al. 1978

Journal of Graph Theory

View full text Add to dashboard Cite

A new upper bound is given for the cycle-complete graph Ramsey number r(C,,,, K"), the smallest order for a graph which forces it to contain either a cycle of order m or a set of ri independent vertices . Then, another cycle-complete graph Ramsey number is studied, namely r( :C,,,, K") the smallest order for a graph which forces it to contain either a cycle of order 1 for some I satisfying 3 :1 !~ ,m or a set of n independent vertices . We obtain the exact value of r( :C,,,, K") for all m > n and an upper bound which applies when m is large in comparison with log n .

show abstract

Section: Notationmentioning

confidence: 90%

On cycle—Complete graph ramsey numbers

Erdös

Faudree

Rousseau

et al. 1978

Journal of Graph Theory

View full text Add to dashboard Cite

show abstract

“…The upper bound actually holds for all k ≥ 2 and ν, and was established by Bondy and Simonovits [3] (see also [14], [40]). The lower bound holds for an infinite sequence of values of ν; c k is a positive function of k only and = 0 if k is odd, and = 1 if k is even.…”

Section: Examples and Applicationsmentioning

confidence: 97%

General properties of some families of graphs defined by systems of equations

Lazebnik

Woldar

2001

Journal of Graph Theory

View full text Add to dashboard Cite

Abstract. In this paper we present a simple method for constructing infinite families of graphs defined by a class of systems of equations over commutative rings. We show that the graphs in all such families possess some general properties including regularity and bi-regularity, existence of special vertex colorings, and existence of covering maps -hence, embedded spectra -between every two members of the same family. Another general property, recently discovered, is that nearly every graph constructed in this manner edge-decomposes either the complete, or complete bipartite, graph which it spans.In many instances, specializations of these constructions have proved useful in various graph theory problems, but especially in many extremal problems. A short survey of the related results is included. We also show that the edgedecomposition property allows one to improve existing lower bounds for some multicolor Ramsey numbers.

show abstract

“…To obtain the results of this section we rely on the following combinatorial lemma of Bondy and Simonovits [BS74].…”

Section: Finding Cycles In Sparse Undirected Graphsmentioning

confidence: 99%