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An integer program and new lower bounds for computing the strong rainbow connection numbers of graphs [article]

Logan A. Smith, David T. Mildebrath, Illya V. Hicks
2021 arXiv   pre-print
We present an integer programming model to compute the strong rainbow connection number, src(G), of any simple graph G.  ...  We demonstrate the efficacy of our methods by computing the strong rainbow connection numbers of graphs containing up to 379 vertices.  ...  The authors would like to thank Zachary Kingston of Rice University for helpful comments and conversations.  ... 
arXiv:2006.02988v2 fatcat:u4aknsbzgfebfebvmshyhafi24

Total $k$-rainbow domination subdivision number in graphs

Rana Khoeilar, Mahla Kheibari, Zehui Shao, Seyed Mahmoud Sheikholeslami
2020 Computer Science Journal of Moldova  
In this paper, we initiate the study of total $k$-rainbow domination subdivision number in graphs and we present sharp bounds for ${\rm sd}_{\gamma_{trk}}(G)$.  ...  In addition, we determine the total 2-rainbow domination subdivision number of complete bipartite graphs and show that the total 2-rainbow domination subdivision number can be arbitrary large.  ...  [2] showed that for any connected graph G of order n ≥ 3, γ trk (G) ≥ ⌈ kn ∆(G)+k−1 ⌉. Applying this lower bound and Corollary 8, the next result follows. Corollary 9.  ... 
doaj:da5600cafd7c4e65bda4e761d81180c3 fatcat:omfptjc6mnedfj3mprh5m3btca

Multicolour paths in graphs: NP-hardness, algorithms, and applications on routing in WDM networks

Rafael F. Santos, Alessandro Andrioni, Andre C. Drummond, Eduardo C. Xavier
2016 Journal of combinatorial optimization  
Additionally, we develop Branch and Bound algorithms, ILPs, and heuristics for the problem. We then perform an experimental analysis of the developed algorithms to compare their performances.  ...  Let G = (V, E) be a (directed) graph with a set of nodes V and a set of edges E in which each edge has an associated positive weight w(i, j), and let C = {1, 2, . . . , x} be a set of x colours, x ∈ N.  ...  Denoted by src(G), the strong rainbow connectivity number of a graph is the minimum number of colours needed to make a graph G strongly rainbow connected.  ... 
doi:10.1007/s10878-016-0003-2 fatcat:dbkuesrlbbhb3mnhrc425bypmm

Super efficiency of efficient geodesics in the complex of curves [article]

Xifeng Jin, William W. Menasco
2021 arXiv   pre-print
The proof relies on a new intersection growth inequality between intersection number of curves and their distance in the complex of curves, together with a thorough analysis of the dot graph associated  ...  and the second author (see arXiv:1408.4133) that there is an explicitly computable list of at most d^(6g-6) candidates for the v_1 vertex.  ...  In section 2, we utilize the intersection number of a minimal filling pair as a lower bound and linear integer programming to establish the growth rate to prove Theorem 1.3.  ... 
arXiv:2008.09665v2 fatcat:cxe3sp75qbckpc7uvecgtemd6e

Minimum color‐degree perfect b ‐matchings

Mariia Anapolska, Christina Büsing, Martin Comis, Tabea Krabs
2020 Networks  
The last speaker of the session will chair the session, with two exceptions for PhD-only sessions: Combinatorial Optimization (Mon 18, Room PP, 16-17) chaired by R.  ...  Schedule The seminar rooms are the Paul Painlevé (PP), the Robert Faure (Z) and the Jean-Baptiste Say (Y) amphitheatres, located in Access 1, lower ground floor.  ...  Remarks We presented a new view of Gomory pure and mixed-integer cuts, aimed at incorporating in the familiar column-generation paradigm, employing the primal simplex algorithm.  ... 
doi:10.1002/net.21974 fatcat:4qgtutmxrrcy7anvxks4fivcii

The b-Chromatic Number of Cubic Graphs

Marko Jakovac, Sandi Klavžar
2010 Graphs and Combinatorics  
We will also report about the status of this problem for graphs with minimum degree at least four. Keywords: Rainbow colouring, rainbow connectivity, extremal problem.  ...  Laborde, Payan and Xuong [1] conjectured that every digraph has an independent set of vertices that meets every longest path. We consider the conjecture for special classes of oriented graphs.  ...  New lower bounds on the independence number of a graph in terms of order, size, and clique number are presented.  ... 
doi:10.1007/s00373-010-0898-9 fatcat:5j4t3ji33bbopoker7veh5aybi

Refuting conjectures in extremal combinatorics via linear programming [article]

Adam Zsolt Wagner
2019 arXiv   pre-print
We apply simple linear programming methods and an LP solver to refute a number of open conjectures in extremal combinatorics.  ...  Acknowledgement: The author is indebted to Bernard Lidický who introduced him to linear programming.  ...  an Integer Program (IP) 1 .  ... 
arXiv:1903.05495v1 fatcat:iuegd3t2krhebhypkvtz7dqy7i

Topological and Geometric Combinatorics

Anders Björner, Gil Kalai, Isabella Novik, Günter Ziegler
2011 Oberwolfach Reports  
We also conjecture a new lower bound on the number of vertices of an even dimensional triangulated manifold in terms of its dimension, connectivity and Euler characteristic.  ...  Some other applications include lower bounds for the connectivity of independence complexes, in particular of claw-free graphs and therefore of matching complexes.  ...  An easy and well-known reduction shows that one need only consider the case where n is a squarefree product of primes.  ... 
doi:10.4171/owr/2011/08 fatcat:kcwryms27fecjjs4hciq6ixa2q

When Maximum Stable Set can be solved in FPT time [article]

Édouard Bonnet, Nicolas Bousquet, Stéphan Thomassé, Rémi Watrigant
2019 arXiv   pre-print
In this paper, we introduce some variants of co-graphs with parameterized noise, that is, graphs that can be made into disjoint unions or complete sums by the removal of a certain number of vertices and  ...  the addition/deletion of a certain number of edges per incident vertex, both controlled by the parameter.  ...  Ramsey numbers For two positive integers a and b, R(a, b) is the smallest integer such that any graph with at least that many vertices has an independent set of size a or a clique of size b.  ... 
arXiv:1909.08426v1 fatcat:zejkso7etzgh5c4qemvfwvfpbm

Algebra in Computational Complexity (Dagstuhl Seminar 14391)

Manindra Agrawal, Valentine Kabanets, Thomas Thierauf, Christopher Umans, Marc Herbstritt
2015 Dagstuhl Reports  
NP and circuit lower bounds, the effort to resolve the complexity of matrix multiplication, and a framework for constructing locally testable codes.  ...  The Razborov-Smolensky polynomial-approximation method for proving constantdepth circuit lower bounds, the PCP characterization of NP, and the Agrawal-Kayal-Saxena polynomial-time primality test are some  ...  the number of integer roots of a univariate polynomial should be polynomially bounded in the size of the smallest straight-line program computing it.  ... 
doi:10.4230/dagrep.4.9.85 dblp:journals/dagstuhl-reports/AgrawalKTU14 fatcat:iailudx2rrhaxenbi3d7ou73ly

Ramsey Theory in the Work of Paul Erdős [chapter]

Ron L. Graham, Jaroslav Nešetřil
2013 The Mathematics of Paul Erdős II  
, in theoretical computer science for lower bounds for various measures of complexity.  ...  For example, the search for an explicit graph of size say 2 n=2 which w ould demonstrate this Ramsey lower bound has been so far unsuccessful.  ... 
doi:10.1007/978-1-4614-7254-4_13 fatcat:vqvyhfhj3ngr7c3vokrsmrt2ai

Ramsey Theory in the Work of Paul Erdős [chapter]

R. L. Graham, J. Nešetřil
1997 Algorithms and Combinatorics  
, in theoretical computer science for lower bounds for various measures of complexity.  ...  For example, the search for an explicit graph of size say 2 n=2 which w ould demonstrate this Ramsey lower bound has been so far unsuccessful.  ... 
doi:10.1007/978-3-642-60406-5_16 fatcat:cqkc3inzjfe4lbu2jj4oem2fv4

Integrality gaps for colorful matchings

Steven Kelk, Georgios Stamoulis
2019 Discrete Optimization  
We study the integrality gap of the natural linear programming relaxation for the Bounded Color Matching (BCM) problem.  ...  We provide several families of instances and establish lower bounds on their integrality gaps and we study how the Sherali-Adams "lift-and-project" technique behaves on these instances.  ...  E k ) be an instance of this rainbow matching problem. For each color class C j , create a new vertex c j and let N be the set of all these new vertices.  ... 
doi:10.1016/j.disopt.2018.12.003 fatcat:64jqlwfopjfrxi4dczlirzqxma

Mathematical and Algorithmic Analysis of Network and Biological Data [article]

Charalampos E. Tsourakakis
2014 arXiv   pre-print
This dissertation contributes to mathematical and algorithmic problems that arise in the analysis of network and biological data.  ...  The rainbow connectivity rc(G) of a connected graph G is the smallest number of colors that are needed in order to make G rainbow edge connected.  ...  Furthermore, we use random graphs to perform an average case analysis for rainbow connectivity, an intriguing connectivity concept.  ... 
arXiv:1407.0375v1 fatcat:6s2qka5fazbl7bzxz4k3u75hzm

Multicolor and directed edit distance

Maria Axenovich, Ryan R. Martin
2011 Journal of Combinatorics  
The theory for computing the edit distance is extended using random structures and so-called types or colored homomorphisms of graphs.  ...  In this paper, a generalization of graph editing is considered for multicolorings of the complete graph as well as for directed graphs.  ...  Let H be a hereditary property of digraphs and P a palette. Fix a density vector with respect to P, p = (p, q). The limit dist(p, H) := lim n→∞ dist n (p, H) exists. Moreover,  ... 
doi:10.4310/joc.2011.v2.n4.a4 fatcat:j4torni2jvhdxduki6xtawk2ke
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