2011/10/13 by Brendan D. McKay, McKay, Brendan D., Pascal Schweitzer +3
Computer Science · Mathematics · #05C75 (Primary) 05C62 (Secondary) #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #math.CO #msc:05C62 #msc:05C75
paper · pdf · doi:10.48550/arxiv.1110.2933
16 pages, 2 figures. v3 contains minor typographic changes. As accepted for SIAM J Disc Math
openalex publication_date 2011/10/13 · arxiv created 2013/10/23 · arxiv updated 2013/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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.