Rad HAZU, Matematičke znanosti, Vol. 22 (2018), 1-22.


Tanja Vojković, Damir Vukičević and Vinko Zlatić

Department of Mathematics, Faculty of Science, University of Split, 21000 Split, Croatia
e-mail: tanja@pmfst.hr

Department of Mathematics, Faculty of Science, University of Split, 21000 Split, Croatia
e-mail: vukicevi@pmfst.hr

Institut Ruđer Bošković, 10 000 Zagreb, Croatia
CRN Institute for Complex Systems, via dei Taurini 19, 00185 Rome, Italy
e-mail: Vinko.Zlatic@irb.hr

Abstract.   Vertex coloring and multicoloring of graphs are a well known subject in graph theory, as well as their applications. In vertex multicoloring, each vertex is assigned some subset of a given set of colors. Here we propose a new kind of vertex multicoloring, motivated by the situation of sharing a secret and securing it from the actions of some number of attackers. We name the multicoloring a highly a-resistant vertex k-multicoloring, where a is the number of the attackers, and k the number of colors. For small values a we determine what is the minimal number of vertices a graph must have in order to allow such a coloring, and what is the minimal number of colors needed.

2010 Mathematics Subject Classification.   05C82, 05C15, 68R10, 94A62.

Key words and phrases.   Graph theory, graph coloring, multicoloring, secret sharing.

Full text (PDF) (free access)

DOI: https://doi.org/10.21857/m3v76t6jky


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