scholarly journals The Rectangle Covering Number of Random Boolean Matrices

10.37236/5576 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Mozhgan Pourmoradnasseri ◽  
Dirk Oliver Theis
Keyword(s):  
Chromatic Number ◽  
Communication Complexity ◽  
Boolean Matrix ◽  
Covering Number ◽  
Fractional Chromatic Number ◽  
Binary Logarithm

The rectangle covering number of an $n$-by-$n$ Boolean matrix $M$ is the smallest number of 1-rectangles which are needed to cover all the 1-entries of $M$. Its binary logarithm is the Nondeterministic Communication Complexity, and it equals the chromatic number of a graph $G(M)$ obtained from $M$ by a construction of Lovasz and Saks.We determine the rectangle covering number and related parameters (clique size, independence ratio, fractional chromatic number of $G(M)$) of random Boolean matrices, where each entry is 1 with probability $p = p(n)$, and the entries are independent.

scholarly journals Cubical coloring — fractional covering by cuts and semidefinite programming

2015 ◽  
Vol Vol. 17 no.2 (Graph Theory) ◽  
Author(s):  
Robert Šámal
Keyword(s):  
Semidefinite Programming ◽  
Chromatic Number ◽  
Fractional Chromatic Number ◽  
Maximum Cut ◽  
Continuous Mappings ◽  
Polynomial Time Approximation Algorithm ◽  
Eigenvalue Bound ◽  
International Audience ◽  
Graph Parameters ◽  
Fractional Covering

International audience We introduce a new graph parameter that measures fractional covering of a graph by cuts. Besides being interesting in its own right, it is useful for study of homomorphisms and tension-continuous mappings. We study the relations with chromatic number, bipartite density, and other graph parameters. We find the value of our parameter for a family of graphs based on hypercubes. These graphs play for our parameter the role that cliques play for the chromatic number and Kneser graphs for the fractional chromatic number. The fact that the defined parameter attains on these graphs the correct value suggests that our definition is a natural one. In the proof we use the eigenvalue bound for maximum cut and a recent result of Engström, Färnqvist, Jonsson, and Thapper [An approximability-related parameter on graphs – properties and applications, DMTCS vol. 17:1, 2015, 33–66]. We also provide a polynomial time approximation algorithm based on semidefinite programming and in particular on vector chromatic number (defined by Karger, Motwani and Sudan [Approximate graph coloring by semidefinite programming, J. ACM 45 (1998), no. 2, 246–265]).

scholarly journals The fractional chromatic number of Zykov products of graphs

2011 ◽  
Vol 24 (4) ◽  
pp. 432-437 ◽  
Author(s):  
Pierre Charbit ◽  
Jean Sébastien Sereni
Keyword(s):  
Chromatic Number ◽  
Fractional Chromatic Number

scholarly journals The Non-Crossing Graph

10.37236/1140 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Nathan Linial ◽  
Michael Saks ◽  
David Statter
Keyword(s):  
Chromatic Number ◽  
Independence Number ◽  
Clique Number ◽  
The Other ◽  
Fractional Chromatic Number ◽  
Vertex Set

Two sets are non-crossing if they are disjoint or one contains the other. The non-crossing graph ${\rm NC}_n$ is the graph whose vertex set is the set of nonempty subsets of $[n]=\{1,\ldots,n\}$ with an edge between any two non-crossing sets. Various facts, some new and some already known, concerning the chromatic number, fractional chromatic number, independence number, clique number and clique cover number of this graph are presented. For the chromatic number of this graph we show: $$ n(\log_e n -\Theta(1)) \le \chi({\rm NC}_n) \le n (\lceil\log_2 n\rceil-1). $$

scholarly journals 1-Subdivisions, the Fractional Chromatic Number and the Hall Ratio

COMBINATORICA ◽  
2020 ◽  
Author(s):  
Zdenĕk Dvořák ◽  
Patrice Ossona de Mendez ◽  
Hehui Wu
Keyword(s):  
Chromatic Number ◽  
Fractional Chromatic Number

scholarly journals Bounding the Fractional Chromatic Number of $K_\Delta$-Free Graphs

2013 ◽  
Vol 27 (2) ◽  
pp. 1184-1208 ◽  
Author(s):  
Katherine Edwards ◽  
Andrew D. King
Keyword(s):  
Chromatic Number ◽  
Fractional Chromatic Number
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