  The competition graph of a directed acyclic graph D is the undirected graph
on the same vertex set as D in which two distinct vertices are adjacent if they
have a common out-neighbor in D. The competition number of an undirected graph
G is the least number of isolated vertices that have to be added to G to make
it the competition graph of a directed acyclic graph. We resolve two
conjectures concerning competition graphs. First we prove a conjecture of Opsut
by showing that the competition number of every quasi-line graph is at most 2.
Recall that a quasi-line graph, also called a locally co-bipartite graph, is a
graph for which the neighborhood of every vertex can partitioned into at most
two cliques. To prove this conjecture we devise an alternative characterization
of quasi-line graphs to the one by Chudnovsky and Seymour. Second, we prove a
conjecture of Kim by showing that the competition number of any graph is at
most one greater than the number of holes in the graph. Our methods also allow
us to prove a strengthened form of this conjecture recently proposed by Kim,
Lee, Park and Sano, showing that the competition number of any graph is at most
one greater than the dimension of the subspace of the cycle space spanned by
the holes.
