On Suborbital Graphs for the Extended Modular Group $${\hat{\Gamma}}$$

Chromatic Numbers of Suborbital Graphs for the Modular Group and the Extended Modular Group

“…Likewise, in [3], the author used the notationĜ , to denote the suborbital graph forΓ onQ, andF , is the subgraph ofĜ , on the orbit (Γ 0 ( ))(∞) whereΓ 0 ( ) is a subgroup ofΓ determined bŷ Γ 0 ( ) = {( ) ∈Γ : ≡ 0 mod } .…”

“…This section provides the vertex coloring results for graphŝ F , andĜ , . Two lemmas below, edge conditions for F , andF , , are the useful results obtained in [1] and [3], respectively. They provide the relation between F , andF , that we can use to extend the results from the previous section.…”

“…In this case, the graph G(V, ) is called a self-paired suborbital graph. Refer to [3,Corollary 3]; the suborbital graphĜ , ofΓ is selfpaired if and only if 2 ≡ ±1 mod . This fact can be applied toF , .…”

“…In 2013, the concept of suborbital graphs continued to the extended modular groupΓ, as in [3]. The notations are defined likewise.…”

Solutions to some congruence equations via suborbital graphs

“…2011 , 2015 ; Kader et al. 2010 ; Kader and Güler 2013 ; Kesicioğlu et al. 2013 ; Keskin 2006 ; Kör et al.…”