Šámal, Robert - Cubical coloring — fractional covering by cuts and semidefinite programming

dmtcs:2134 - Discrete Mathematics & Theoretical Computer Science, November 18, 2015, Vol. 17 no.2
Cubical coloring — fractional covering by cuts and semidefinite programming

Authors: Šámal, Robert

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]).

Source : oai:HAL:hal-01349049v1
Volume: Vol. 17 no.2
Section: Graph Theory
Published on: November 18, 2015
Submitted on: July 31, 2012
Keywords: covering,graph coloring,maxcut,cut-continuous maps,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]


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