Highly resistant multicoloring with 3 attackers and 1 malfunctioning vertex
In this paper we explore a way of securing a secret inside a graph by observing pieces of the secret as colors assigned to the graph vertices. If a graph allows a highly (a, b)-resistant k-multicoloring then a secret can be divided into k parts and sets of those parts distributed to the vertices of the graph so that no a attackers can steal the secret, and when a attackers and b malfuntioning vertices leave the graph, the secret is still whole in the remaining graph. In this paper we explore how many vertices a graph must have in order to allow a highly (3, 1)-resistant k-multicoloring, and what is the minimal number of colors, for graphs that do allow such multicoloring.