Minimally 4-restricted edge connected graphs

Kamyab, Khalid (Department of Mathematics‎, ‎Urmia University‎, ‎Urmia 57135‎, ‎Iran)
Ghasemi, Mohsen (Department of Mathematics‎, ‎Urmia University‎, ‎Urmia 57135‎, ‎Iran)
Varmazyar, Rezvan (Department of Mathematics‎, ‎Khoy Branch‎, ‎Azad University‎, ‎Khoy‎, ‎Iran)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 13, No 2 (2025): Electronic Journal of Graph Theory and Applications

Suppose that G is minimally 4-restricted edge connected graph without triangle, δ(G) ≥ 2 and α4(G) ≥ 6. Suppose that A is a λ4-atom of G such that for each path of length 3 in A, say P3 = xyz, we have dA(y) − 1 = dA−{x,y,z}(y), where x ∼ y ∼ z. In this paper, under these assumptions, we show that all atoms of G are trivial.