A generalization of Opsut’s result on the competition numbers of line graphs

In this paper, we prove that if a graph G is diamond-free, then the competition number of G is bounded above by 2+12∑v∈Vh(G)(θV(NG(v))−2) where Vh(G) is the set of nonsimplicial vertices of G. This result generalizes Opsut’s result for line graphs. We also show that the upper bound holds for certain...

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Bibliographic Details
Published inDiscrete Applied Mathematics Vol. 181; pp. 152 - 159
Main Authors Kim, Suh-Ryung, Lee, Jung Yeun, Park, Boram, Sano, Yoshio
Format Journal Article
LanguageEnglish
Published Elsevier B.V 30.01.2015
Subjects
Competition
Competition graph
Competition number
Diamond-free graph
Diamonds
Edge clique cover
Edge-buffered graph
Graph theory
Graphs
Line graph
Mathematical analysis
Opsut’s conjecture
Quasi-line graph
Upper bounds
Opsut’s conjecture
Diamond-free graph
Competition number
Line graph
Competition graph
Quasi-line graph
Edge-buffered graph
Edge clique cover
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Summary:In this paper, we prove that if a graph G is diamond-free, then the competition number of G is bounded above by 2+12∑v∈Vh(G)(θV(NG(v))−2) where Vh(G) is the set of nonsimplicial vertices of G. This result generalizes Opsut’s result for line graphs. We also show that the upper bound holds for certain graphs which might have diamonds. As a matter of fact, we go further to a conjecture that the above upper bound holds for the competition number of any graph, which leads to a natural generalization of Opsut’s conjecture.
Bibliography:ObjectType-Article-1
SourceType-Scholarly Journals-1
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content type line 23
ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2014.10.014