A Generalization of Opsut’s Lower Bounds for the Competition Number of a Graph

• Published in: Graphs and combinatorics Vol. 29; no. 5; pp. 1543 - 1547
• Main Author: Sano, Yoshio
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01.09.2013; Springer Nature B.V
• Subjects: Combinatorial analysis; Combinatorics; Competition; Engineering Design; Graph theory; Graphs; Lower bounds; Mathematics; Mathematics and Statistics; Original Paper; 05C20; Vertex clique cover number; Competition number; 05C69; Competition graph; Edge clique cover number
• ISSN: 0911-0119; 1435-5914
• DOI: 10.1007/s00373-012-1188-5

The notion of a competition graph was introduced by Cohen in 1968. The competition graph C ( D ) of a digraph D is a (simple undirected) graph which has the same vertex set as D and has an edge between two distinct vertices x and y if and only if there exists a vertex v in D such that ( x , v ) and ( y , v ) are arcs of D . For any graph G , G together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. In 1978, Roberts defined the competition number k ( G ) of a graph G as the minimum number of such isolated vertices. In general, it is hard to compute the competition number k ( G ) for a graph G and it has been one of the important research problems in the study of competition graphs to characterize a graph by its competition number. In 1982, Opsut gave two lower bounds for the competition number of a graph. In this paper, we give a generalization of these two lower bounds for the competition number of a graph.