Some New Results in the Alcuin Number of Graphs

TL;DR

This paper investigates the Alcuin number of graphs, providing classifications and conditions for certain graph classes, and analyzing properties of graph products and regular graphs to advance understanding of graph cover problems.

Contribution

It introduces a classification of graphs with unique minimum vertex cover and establishes necessary and sufficient conditions for class two graphs, enhancing the theoretical framework of graph cover analysis.

Findings

Classification of graphs with unique minimum vertex cover

Necessary and sufficient conditions for class two graphs

Results on cartesian product and regular 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.