On p-competition graphs of loopless Hamiltonian digraphs without symmetric arcs and graph operations

Authors

Kuniharu Yokomura, Tokai UniversityFollow
Morimasa Tsuchiya, Tokai UniversityFollow

Publication Date

July 2022

Abstract

For a digraph D, the p-competition graph C_p(D) of D is the graph satisfying the following: V(C_p(D))=V(D), for x,y ∈ V(C_p(D)), xy ∈ E(C_p(D)) if and only if there exist distinct p vertices v₁, v₂, ..., v_p ∈ V(D) such that x → v_i, y → v_i ∈ A(D) for each i=1,2, ..., p.

We show the H₁ ∪ H₂ is a p-competition graph of a loopless digraph without symmetric arcs for p ≥ 2 , where H₁ and H₂ are p-competition graphs of loopless digraphs without symmetric arcs and V(H₁) ∩ V(H₂) = {α}. For p-competition graphs of loopless Hamiltonian digraphs without symmetric arcs, we obtain similar results. And we show that a star K_1,n is a p-competition graph of a loopless Hamiltonian digraph without symmetric arcs if n ≥ 2p+3 and p ≥ 3 .

Based on these results, we obtain conditions such that spiders, caterpillars and cacti are p-competition graphs of loopless digraphs without symmetric arcs. We also obtain conditions such that these graphs are p-competition graphs of loopless Hamiltonian digraphs without symmetric arcs.

Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

Recommended Citation

Yokomura, Kuniharu and Tsuchiya, Morimasa (2022) "On p-competition graphs of loopless Hamiltonian digraphs without symmetric arcs and graph operations," Theory and Applications of Graphs: Vol. 9: Iss. 2, Article 3.
DOI: 10.20429/tag.2022.090203
Available at: https://digitalcommons.georgiasouthern.edu/tag/vol9/iss2/3