scholarly journals Large Cuts with Local Algorithms on Triangle-Free Graphs

10.37236/6862 ◽  
2017 ◽  
Vol 24 (4) ◽  
Author(s):  
Juho Hirvonen ◽  
Joel Rybicki ◽  
Stefan Schmid ◽  
Jukka Suomela
Keyword(s):  
Lower Bound ◽  
Local Algorithm ◽  
Computational Techniques ◽  
Free Graph ◽  
Local Algorithms ◽  
Bipartite Subgraph ◽  
Classic Result ◽  
Maximum Cut ◽  
Regular Triangle ◽  
Neighbourhood Graph

Let $G$ be a $d$-regular triangle-free graph with $m$ edges. We present an algorithm which finds a cut in $G$ with at least $(1/2 + 0.28125/\sqrt{d})m$ edges in expectation, improving upon Shearer's classic result. In particular, this implies that any $d$-regular triangle-free graph has a cut of at least this size, and thus, we obtain a new lower bound for the maximum number of edges in a bipartite subgraph of $G$.Our algorithm is simpler than Shearer's classic algorithm and it can be interpreted as a very efficient randomised distributed (local) algorithm: each node needs to produce only one random bit, and the algorithm runs in one round. The randomised algorithm itself was discovered using computational techniques. We show that for any fixed $d$, there exists a weighted neighbourhood graph $\mathcal{N}_d$ such that there is a one-to-one correspondence between heavy cuts of $\mathcal{N}_d$ and randomised local algorithms that find large cuts in any $d$-regular input graph. This turns out to be a useful tool for analysing the existence of cuts in $d$-regular graphs: we can compute the optimal cut of $\mathcal{N}_d$ to attain a lower bound on the maximum cut size of any $d$-regular triangle-free graph.

scholarly journals Lower Bounds for the Size of Random Maximal $H$-Free Graphs

10.37236/93 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Guy Wolfovitz
Keyword(s):  
Lower Bound ◽  
Random Process ◽  
Lower Bounds ◽  
Random Permutation ◽  
Bipartite Graphs ◽  
Expected Number ◽  
Free Graph ◽  
Vertex Graph

We consider the next random process for generating a maximal $H$-free graph: Given a fixed graph $H$ and an integer $n$, start by taking a uniformly random permutation of the edges of the complete $n$-vertex graph $K_n$. Then, traverse the edges of $K_n$ according to the order imposed by the permutation and add each traversed edge to an (initially empty) evolving $n$-vertex graph - unless its addition creates a copy of $H$. The result of this process is a maximal $H$-free graph ${\Bbb M}_n(H)$. Our main result is a new lower bound on the expected number of edges in ${\Bbb M}_n(H)$, for $H$ that is regular, strictly $2$-balanced. As a corollary, we obtain new lower bounds for Turán numbers of complete, balanced bipartite graphs. Namely, for fixed $r \ge 5$, we show that ex$(n, K_{r,r}) = \Omega(n^{2-2/(r+1)}(\ln\ln n)^{1/(r^2-1)})$. This improves an old lower bound of Erdős and Spencer. Our result relies on giving a non-trivial lower bound on the probability that a given edge is included in ${\Bbb M}_n(H)$, conditioned on the event that the edge is traversed relatively (but not trivially) early during the process.

scholarly journals On subgraphs of C2k-free graphs and a problem of Kühn and Osthus

2020 ◽  
Vol 29 (3) ◽  
pp. 436-454
Author(s):  
Dániel Grósz ◽  
Abhishek Methuku ◽  
Casey Tompkins
Keyword(s):  
Short Proof ◽  
Uniform Hypergraph ◽  
Free Graph ◽  
Bipartite Subgraph ◽  
Image Position

AbstractLet c denote the largest constant such that every C6-free graph G contains a bipartite and C4-free subgraph having a fraction c of edges of G. Győri, Kensell and Tompkins showed that 3/8 ⩽ c ⩽ 2/5. We prove that c = 38. More generally, we show that for any ε > 0, and any integer k ⩾ 2, there is a C2k-free graph $G'$ which does not contain a bipartite subgraph of girth greater than 2k with more than a fraction $$\Bigl(1-\frac{1}{2^{2k-2}}\Bigr)\frac{2}{2k-1}(1+\varepsilon)$$ of the edges of $G'$ . There also exists a C2k-free graph $G''$ which does not contain a bipartite and C4-free subgraph with more than a fraction $$\Bigl(1-\frac{1}{2^{k-1}}\Bigr)\frac{1}{k-1}(1+\varepsilon)$$ of the edges of $G''$ .One of our proofs uses the following statement, which we prove using probabilistic ideas, generalizing a theorem of Erdős. For any ε > 0, and any integers a, b, k ⩾ 2, there exists an a-uniform hypergraph H of girth greater than k which does not contain any b-colourable subhypergraph with more than a fraction $$\Bigl(1-\frac{1}{b^{a-1}}\Bigr)(1+\varepsilon)$$ of the hyperedges of H. We also prove further generalizations of this theorem.In addition, we give a new and very short proof of a result of Kühn and Osthus, which states that every bipartite C2k-free graph G contains a C4-free subgraph with at least a fraction 1/(k−1) of the edges of G. We also answer a question of Kühn and Osthus about C2k-free graphs obtained by pasting together C2l’s (with k >l ⩾ 3).

BIPARTITE SUBGRAPHS OF -FREE GRAPHS

2017 ◽  
Vol 96 (1) ◽  
pp. 1-13 ◽  
Author(s):  
QINGHOU ZENG ◽  
JIANFENG HOU
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Maximum Degree ◽  
Bipartite Subgraph

For a graph $G$, let $f(G)$ denote the maximum number of edges in a bipartite subgraph of $G$. For an integer $m$ and for a fixed graph $H$, let $f(m,H)$ denote the minimum possible cardinality of $f(G)$ as $G$ ranges over all graphs on $m$ edges that contain no copy of $H$. We give a general lower bound for $f(m,H)$ which extends a result of Erdős and Lovász and we study this function for any bipartite graph $H$ with maximum degree at most $t\geq 2$ on one side.

scholarly journals On the sum-connectivity index

Filomat ◽  
2011 ◽  
Vol 25 (3) ◽  
pp. 29-42 ◽  
Author(s):  
Shilin Wang ◽  
Zhou Bo ◽  
Nenad Trinajstic
Keyword(s):  
Lower Bound ◽  
Minimum Degree ◽  
Simple Graph ◽  
Connectivity Index ◽  
Extremal Graphs ◽  
Free Graph ◽  
Mathematical Chemistry ◽  
Degree Of Vertex

The sum-connectivity index of a simple graph G is defined in mathematical chemistry as R+(G) = ? uv?E(G)(du+dv)?1/2, where E(G) is the edge set of G and du is the degree of vertex u in G. We give a best possible lower bound for the sum-connectivity index of a graph (a triangle-free graph, respectively) with n vertices and minimum degree at least two and characterize the extremal graphs, where n ? 11.

scholarly journals New Computational Upper Bounds for Ramsey Numbers $R(3,k)$

10.37236/2824 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Jan Goedgebeur ◽  
Stanisław P. Radziszowski
Keyword(s):  
Lower Bounds ◽  
Independent Sets ◽  
Upper Bounds ◽  
Computational Techniques ◽  
Free Graph ◽  
Ramsey Numbers ◽  
Critical Graph ◽  
Minimum Number

Using computational techniques we derive six new upper bounds on the classical two-color Ramsey numbers: $R(3,10) \le 42$, $R(3,11) \le 50$, $R(3,13) \le 68$, $R(3,14) \le 77$, $R(3,15) \le 87$, and $R(3,16) \le 98$. All of them are improvements by one over the previously best known bounds. Let $e(3,k,n)$ denote the minimum number of edges in any triangle-free graph on $n$ vertices without independent sets of order $k$. The new upper bounds on $R(3,k)$ are obtained by completing the computation of the exact values of $e(3,k,n)$ for all $n$ with $k \leq 9$ and for all $n \leq 33$ for $k = 10$, and by establishing new lower bounds on $e(3,k,n)$ for most of the open cases for $10 \le k \le 15$. The enumeration of all graphs witnessing the values of $e(3,k,n)$ is completed for all cases with $k \le 9$. We prove that the known critical graph for $R(3,9)$ on 35 vertices is unique up to isomorphism. For the case of $R(3,10)$, first we establish that $R(3,10)=43$ if and only if $e(3,10,42)=189$, or equivalently, that if $R(3,10)=43$ then every critical graph is regular of degree 9. Then, using computations, we disprove the existence of the latter, and thus show that $R(3,10) \le 42$.

scholarly journals Clique-transversal sets and weak 2-colorings in graphs of small maximum degree

2009 ◽  
Vol Vol. 11 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Gábor Bacsó ◽  
Zsolt Tuza
Keyword(s):  
Lower Bound ◽  
Polynomial Time ◽  
Connected Graph ◽  
Maximum Degree ◽  
Free Graph ◽  
Polynomial Time Algorithms ◽  
Odd Cycle ◽  
Vertex Set ◽  
International Audience

Graphs and Algorithms International audience A clique-transversal set in a graph is a subset of the vertices that meets all maximal complete subgraphs on at least two vertices. We prove that every connected graph of order n and maximum degree three has a clique-transversal set of size left perpendicular19n/30 + 2/15right perpendicular. This bound is tight, since 19n/30 - 1/15 is a lower bound for infinitely many values of n. We also prove that the vertex set of any connected claw-free graph of maximum degree at most four, other than an odd cycle longer than three, can be partitioned into two clique-transversal sets. The proofs of both results yield polynomial-time algorithms that find corresponding solutions.

scholarly journals Multi-crossing number for knots and the Kauffman bracket polynomial

2016 ◽  
Vol 164 (1) ◽  
pp. 147-178 ◽  
Author(s):  
COLIN ADAMS ◽  
ORSOLA CAPOVILLA-SEARLE ◽  
JESSE FREEMAN ◽  
DANIEL IRVINE ◽  
SAMANTHA PETTI ◽  
...  
Keyword(s):  
Singular Point ◽  
Lower Bound ◽  
Crossing Number ◽  
Crossing Numbers ◽  
Kauffman Bracket ◽  
Classic Result ◽  
Formula Group ◽  
Image Position ◽  
Bracket Polynomial ◽  
Extensive List

AbstractA multi-crossing (or n-crossing) is a singular point in a projection of a knot or link at which n strands cross so that each strand bisects the crossing. We generalise the classic result of Kauffman, Murasugi and Thistlethwaite relating the span of the bracket polynomial to the double-crossing number of a link, span〈K〉 ⩽ 4c2, to the n-crossing number. We find the following lower bound on the n-crossing number in terms of the span of the bracket polynomial for any n ⩾ 3: $$\text{span} \langle K \rangle \leq \left(\left\lfloor\frac{n^2}{2}\right\rfloor + 4n - 8\right) c_n(K).$$ We also explore n-crossing additivity under composition, and find that for n ⩾ 4 there are examples of knots K1 and K2 such that cn(K1#K2) = cn(K1) + cn(K2) − 1. Further, we present the the first extensive list of calculations of n-crossing numbers of knots. Finally, we explore the monotonicity of the sequence of n-crossings of a knot, which we call the crossing spectrum.

scholarly journals Local Computation of Simultaneous Fixed-Points

2003 ◽  
Vol 21 (420) ◽  
Author(s):  
Henrik Reif Andersen
Keyword(s):  
Model Checking ◽  
Fixed Points ◽  
Ad Hoc ◽  
General Algorithm ◽  
General Mechanism ◽  
Local Algorithm ◽  
Worst Case ◽  
Local Algorithms ◽  
Logarithmic Factor ◽  
Strictness Analysis

<!--DAIMI Standard Header Begin--> <p>We present a very simple, yet general algorithm for computing simultaneous, minimum fixed-points of monotonic functions, or turning the newpoint slightly, an algorithm for computing minimum solutions to a system of monotonic equations. The algorithm is local (demand-driven, lazy, ... ), i.e. it will try to determine the value of a single component in the simultaneous fixed-point by investigating only certain necessary parts of the description of the monotonic function, or in terms of the equational presentation, it will determine the value of a single variable by investigating only a part of the equational system.</p><p>In the worst-case this involves inspecting the complete system, and the algorithm will be a logarithmic factor worse than a global algorithm (computing the values of all variables simultaneously). But despite its simplicity the local algorithm has some advantages which promise much better performance on typical cases. The algorithm should be seen as a schema that for any particular application needs to be refined to achieve better efficiency, but the general mechanism remains the same. As such it seems to achieve performance comparable to, and for some examples improving upon, carefully designed <em>ad hoc</em> algorithms, still maintaining the benefits of being local.</p><p>We illustrate this point by tailoring the general algorithm to concrete examples in such (apparently) diverse areas as type inference, model checking, and strictness analysis. Especially in connection with the last example, strictness analysis, and more generally abstract interpretation, it is illustrated how the local algorithm provides a very minimal approach when determining the fixed-points, reminiscent of, but improving upon, what is known as Pending Analysis. In the case of model checking a specialised version of the algorithm has already improved on earlier known local algorithms.</p>

Linear-Time Approximation Algorithms for the Max Cut Problem

1993 ◽  
Vol 2 (2) ◽  
pp. 201-210 ◽  
Author(s):  
Nguyen van Ngoc ◽  
Zsolt Tuza
Keyword(s):  
Lower Bound ◽  
Approximation Algorithms ◽  
Lower Bounds ◽  
Connected Graph ◽  
Linear Time ◽  
Worst Case ◽  
Time Approximation ◽  
Bipartite Subgraph ◽  
Max Cut Problem ◽  
Multiple Edges

Let G be a connected graph with n vertices and m edges (multiple edges allowed), and let k ≥ 2 be an integer. There is an algorithm with (optimal) running time of O(m) that finds(i) a bipartite subgraph of G with ≥ m/2 + (n − 1)/4 edges,(ii) a bipartite subgraph of G with ≥ m/2 + 3(n−1)/8 edges if G is triangle-free,(iii) a k-colourable subgraph of G with ≥ m − m/k + (n−1)/k + (k − 3)/2 edges if k ≥ 3 and G is not k-colorable.Infinite families of graphs show that each of those lower bounds on the worst-case performance are best possible (for every algorithm). Moreover, even if short cycles are excluded, the general lower bound of m − m/k cannot be replaced by m − m/k + εm for any fixed ε > 0; and it is NP-complete to decide whether a graph with m edges contains a k-colorable subgraph with more than m − m/k + εm edges, for any k ≥ 2 and ε> 0, ε < 1/k.

Remote sensing of primary production in the ocean: promise and fulfilment

1995 ◽  
Vol 348 (1324) ◽  
pp. 191-202 ◽  
Keyword(s):  
Remote Sensing ◽  
Primary Production ◽  
North Atlantic ◽  
Local Algorithm ◽  
Ocean Colour ◽  
Remotely Sensed Data ◽  
Local Algorithms ◽  
The North ◽  
Basin Scale ◽  
The North Atlantic

Remote sensing of ocean colour affords us our only window into the synoptic state of the pelagic ecosystem, and is likely to remain the only such option into the foreseeable future. Estimation of primary production from remotely sensed data on ocean colour is a research problem in two parts: (i) the construction of a local algorithm; and (ii) the development of a protocol for extrapolation. Good local algorithms exist but their proper implementation requires that certain parameters be specified. Protocols for extrapolation have to include procedures for the assignment of these parameters. One suitable approach is based on partition of the ocean into a suite of domains and provinces within which physical forcing, and the algal response to it, are distinct. This approach is still in its infancy, but is best developed for the North Atlantic. Using this method, and using the accumulated data from oceanographic expeditions, leads to an estimate for the annual primary production of the North Atlantic at the basin scale. Direct validation of the result is not possible in the absence of an independent calculation, but the potential errors involved may be assessed.

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