Rank and chromatic number of a graph

Kotlov, Andrei

doi:10.1002/(sici)1097-0118(199709)26:1<1::aid-jgt1>3.3.co;2-u

Cited by 9 publications

(13 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since rank(K ) is even, rank(K ) 2ρ − 4. By (11), we have |P 1 ∩ V (K )| (ρ − 2)/2 and therefore, using (7) and Theorem 2.6, we obtain that (ρ − 2)/2 + 3 · 2 ρ−3 − 3 |V (K )| b(2ρ − 4) that is a contradiction to ρ 5. This establishes the desired property of K. Working toward a contradiction, suppose that t 2 = 1.…”

Section: Triangle-free Graphsmentioning

confidence: 91%

“…Proof. If H is an induced subgraph of G with the maximum possible order subject to rank(H ) < rank(G), then the statements (i)-(iv) can be found among the results of [7] and also [8]. In order to prove the rest of the assertion, we let H be an induced subgraph of G with the maximum possible order subject to rank(H ) rank(G) − 2.…”

Section: Lemma 22mentioning

confidence: 97%

“…By the induction hypothesis, K is isomorphic to either C r−2 or C r−3 and so α(K) = 3 · 2 ρ−3 − 1 that contradicts s = 3 · 2 ρ−3 . Hence (7) yields that s = 3 · 2 ρ−3 − 1 and = 1. Then from n c(r), we have p ρ which in turn by (8) gives p = ρ and so k = s + p = c(r − 2).…”

Section: Triangle-free Graphsmentioning

confidence: 98%

“…For a graph G, a subset S of V (G) with |S| > 1 is called a duplication class of G if N(u) = N(v), for every u, v ∈ S. For a subset X of V (G), the notation G − X represents the subgraph obtained by removing the vertices in X from G. Lemma 2.1. [7,8] For any reduced graph G, the following hold.…”

Section: Preliminariesmentioning

confidence: 99%

“…Therefore p 1 = ρ + 1, p 2 = 0 and = 1. Note that from (7) and k = s + p 1 c(r − 2), we have s = 3 · 2 ρ−3 − 2 and thus k = c(r − 2). By the preceding paragraph, K is not bipartite, since otherwise G would be bipartite.…”

Section: Triangle-free Graphsmentioning

confidence: 99%

See 4 more Smart Citations

Maximum Order of Triangle‐Free Graphs with a Given Rank

Ghorbani

Mohammadian²,

Tayfeh-Rezaie³

2014

Journal of Graph Theory

View full text Add to dashboard Cite

show abstract

Section: Triangle-free Graphsmentioning

confidence: 91%

Section: Lemma 22mentioning

confidence: 97%

Section: Triangle-free Graphsmentioning

confidence: 98%

Section: Preliminariesmentioning

confidence: 99%

Section: Triangle-free Graphsmentioning

confidence: 99%

See 3 more Smart Citations

Maximum Order of Triangle‐Free Graphs with a Given Rank

Ghorbani

Mohammadian²,

Tayfeh-Rezaie³

2014

Journal of Graph Theory

View full text Add to dashboard Cite

show abstract

Computational complexity

A Gentle Introduction to Optimization

View full text Add to dashboard Cite

Computational Complexity Theory is the field that studies the inherent costs of algorithms for solving mathematical problems. Its major goal is to identify the limits of what is efficiently computable in natural computational models. Computational complexity ranges from quantum computing to determining the minimum size of circuits that compute basic mathematical functions to the foundations of cryptography and security. Computational complexity emerged from the combination of logic, combinatorics, information theory, and operations research. It coalesced around the central problem of "P versus NP" (one of the seven open problems of the Clay Institute). While this problem remains open, the field has grown both in scope and sophistication. Currently, some of the most active research areas in computational complexity are: 2 Communication Complexity Communication complexity (first introduced by Yao) has proven to be one of the most fruitful areas of complexity theory. In this model, two or more parties each know of a different input, and wish to communicate some joint function of all of their inputs. The complexity of the function is the number of bits they must exchange in order to determine the value of the function. This model has proven to be extremely useful because on the one hand, the model is very simple, and so many computational processes (such as circuits, streaming computation and many others) can be viewed as executing communication protocols. On the other hand, over the years we have found powerful techniques that can be used to prove lowerbounds on the communication complexity of different types of functions, and so obtain lowerbounds on other computational processes. At this workshop, we discussed a few fundamental recent results about communication complexity. 2.1 Presentation Highlights 2.1.1 Information Complexity Mark Braverman gave an overview of information complexity. Information complexity is closely related to communication complexity, but instead of considering the number of bits exchanged, it studies the amount of information (in Shannon's Information Theory sense) that the parties must exchange to solve the problem. Information complexity has found many applications within communication complexity and related areas. The information complexity of problems has been shown to demonstrate some of the attractive properties of Shannon's entropy. For example, it is additive over independent problems. This additivity allows one to obtain, among other things, tight bounds on the communication complexity of problems. In the talk, Mark discussed the information complexity of the AND function (where Alice and Bob each get a bit and need to output the AND of their inputs). Through known connection, understanding the information complexity of the AND function allows one to get a tight formula of 0.4827. .. n + o(n) for the communication complexity of the Set Disjointness problem: where Alice and Bob are given subsets X and Y of {1,. .. , n} and need to decide whether X ∩ Y = ∅. The talk was largely based...

show abstract

Fourier Sparsity, Spectral Norm, and the Log-Rank Conjecture

Tsang

Wong

Xie

et al. 2013

2013 IEEE 54th Annual Symposium on Foundations of Computer Science

View full text Add to dashboard Cite

We study Boolean functions with sparse Fourier spectrum or small spectral norm, and show their applications to the Log-rank Conjecture for XOR functions f (x ⊕ y) -a fairly large class of functions including well studied ones such as Equality and Hamming Distance. The rank of the communication matrix M f for such functions is exactly the Fourier sparsity of f . Let d = deg 2 (f ) be the F2-degree of f and D CC (f • ⊕) stand for the deterministic communication complexity for f (x ⊕ y). We show thatThis improves the (trivial) linear bound by nearly a quadratic factor. We obtain our results through a degree-reduction protocol based on a variant of polynomial rank, and actually conjecture that the communication cost of our protocol is at most log O(1) rank(M f ). The above bounds are obtained from different analysis for the number of parity queries required to reduce f 's F2-degree. Our bounds also hold for the parity decision tree complexity of f , a measure that is no less than the communication complexity.Along the way we also prove several structural results about Boolean functions with small Fourier sparsity f 0 or spectral norm f 1, which could be of independent interest. For functions f with constant F2-degree, we show that: 1) f can be written as the summation of quasi-polynomially many indicator functions of subspaces with ±-signs, improving the previous doubly exponential upper bound by Green and Sanders; 2) being sparse in Fourier domain is polynomially equivalent to having a small parity decision tree complexity; and 3) f depends only on polylog f 1 linear functions of input variables. For functions f with small spectral norm, we show that: 1) there is an affine subspace of co-dimension O( f 1) on which f (x) is a constant, and 2) there is a parity decision tree of depth O( f 1 log f 0) for computing f .

show abstract

Rank and chromatic number of a graph

Cited by 9 publications

References 0 publications

Maximum Order of Triangle‐Free Graphs with a Given Rank

Maximum Order of Triangle‐Free Graphs with a Given Rank

Computational complexity

Fourier Sparsity, Spectral Norm, and the Log-Rank Conjecture

Contact Info

Product

Resources

About