Graphs having many holes but with small competition numbers

• Published in: Applied mathematics letters Vol. 24; no. 8; pp. 1331 - 1335
• Main Authors: Lee, JungYeun, Kim, Suh-Ryung, Kim, Seog-Jin, Sano, Yoshio
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01-08-2011; Elsevier
• Subjects: Clique; Competition; Competition graph; Competition number; Exact sciences and technology; Graph theory; Graphs; Hole; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Sciences and techniques of general use; Clique; Hole; Competition number; Competition graph; Complete graph; Applied mathematics; Connected graph; Bipartite graph; Acyclic graph; Directed graph
• ISSN: 0893-9659; 1873-5452
• DOI: 10.1016/j.aml.2011.03.003

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
K
⌊
n
2
⌋
,
⌈
n
2
⌉
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
ω
=
ω
(
G
)
of
G
satisfies
2
≤
ω
≤
h
+
1
, then the competition number of
G
is at most
h
−
ω
+
3
.