The competition number of a generalized line graph is at most two

TL;DR

This paper extends Opsut's 1982 result by proving that the competition number of generalized line graphs is at most two, and provides conditions for when it equals one, broadening understanding of graph competition numbers.

Contribution

It generalizes Opsut's result from line graphs to generalized line graphs, establishing an upper bound and criteria for the competition number being one.

Findings

Competition number of generalized line graphs is at most two.

Necessary and sufficient conditions for the competition number being one.

Extension of Opsut's 1982 result to a broader class of graphs.

Abstract

In 1982, Opsut showed that the competition number of a line graph is at most two and gave a necessary and sufficient condition for the competition number of a line graph being one. In this note, we generalize this result to the competition numbers of generalized line graphs, that is, we show that the competition number of a generalized line graph is at most two, and give necessary conditions and sufficient conditions for the competition number of a generalized line graph being one.