Some New Results in the Alcuin Number of Graphs

Abstract

We prove some results concerning Alcuin number of graphs. First, we classify graphs which have unique minimum vertex cover. Then we present two necessary conditions for a graph to be of class two and show why one of them (condition on common neighbors) is sufficient as well. By using this classification theorem, we prove some results about class one and class two graphs such as cartesian product of graphs and classification of regular graphs.