Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
1708479 | Applied Mathematics Letters | 2011 | 5 Pages |
Abstract
The competition number k(G) of a graph G is the smallest number k such that G together with k isolated vertices added is the competition graph of an acyclic digraph. A chordless cycle of length at least 4 of a graph is called a hole of the graph. The number of holes of a graph is closely related to its competition number as the competition number of a graph which does not contain a hole is at most one and the competition number of a complete bipartite graph Kân2â,ân2â which has so many holes that no more holes can be added is the largest among those of graphs with n vertices. In this paper, we show that even if a connected graph G has many holes, the competition number of G can be as small as 2 under some assumption. In addition, we show that, for a connected graph G with exactly h holes and at most one non-edge maximal clique, if all the holes of G are pairwise edge-disjoint and the clique number Ï=Ï(G) of G satisfies 2â¤Ïâ¤h+1, then the competition number of G is at most hâÏ+3.
Related Topics
Physical Sciences and Engineering
Engineering
Computational Mechanics
Authors
JungYeun Lee, Suh-Ryung Kim, Seog-Jin Kim, Yoshio Sano,