Minimally 4-restricted edge connected graphs
Abstract
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.
Keywords
Full Text:
PDFDOI: http://dx.doi.org/10.5614/ejgta.2025.13.2.10
References
M. Bai, Y. Tian, and J. Yin, The super restricted edge-connectedness of direct product graphs, Parallel Process. Lett. 33(3) (2023) 2350008:1–2350008:7.
H. Cheng, R. Varmazyar, and M. Ghasemi, On k-restricted connectivity of direct product of graphs, Discrete Math. Algorithms Appl. 15(8) (2023) 2250175.
M. Ghasemi, Some results about the reliability of folded hypercubes, Bull. Malaysian Math. Sci. Soc. 44 (2021) 1093–1099.
M. Ghasemi, On the reliability of modified bubble-sort graphs, Trans. Combinatorics. 11(2) (2022) 77–83.
A. Hellwig and L. Volkmann, Maximally edge-connected and vertex-connected graphs and digraphs: A survey, Discrete Math. 308 (2008) 3265–3296.
Y.M. Hong, Q.H. Liu, and Z. Zhang, Minimally restricted edge connected graphs, App. Math. Lett. 21 (2008) 820–823.
Q.G. Liu, Y.M. Hong, and Z. Zhang, Minimally 3-restricted edge connected graphs, Discrete Appl. Math. 157 (2009) 685–690.
X.M. Liu and J. Meng, The k-restricted edge-connectivity of the data center network DCell, Appl. Math. Comput. 396 (2021) 125941.
S. Liu, C. Ouyang, and J. Ou, Sufficient conditions for optimally and super m-restricted edge-connected graphs with given girth, Linear Multilinear Algebra. 70(6) (2020) 1146–1158.
L. Lovász, Combinatorial Problems and Exercises, North-Holland Publishing Company, 1979.
T. Ma, J. Wang, and M. Zhang, The restricted edge-connectivity of kronecker product graphs, Parallel Process. Lett. 29(3) (2019) 1950012:1–1950012:7.
J.X. Meng, Optimally super-edge-connected transitive graphs, Discrete Math. 260 (2003) 239–248.
H. Tarakmi, H. Azanchilar, M. Ghasemi, and Gh. Azadi, n-restricted edge connectivity of m-barrel fullerene graphs, Iran J. Sci. Technol Trans. A. Sci. 45 (2021) 997–1004.
Y.Q. Wang and Q. Li, Super-edge-connectivity properties of graphs with diameter 2, J. Shanghai Jiaotong Univ. 33(6) (1999) 647–649 (in Chinese).
Y.Q. Wang and Q. Li, Sufficient conditions for a graph to be maximally restricted edge-connected, J. Shanghai Univ. 35(8) (2001) 1253–1255 (in Chinese).
J.M. Xu and K.L. Xu, On restricted edge-connectivity of graphs, Discrete Math. 243 (2002) 291–298.
Z. Zhang and J.X. Meng, Restricted edge connectivity of edge transitive graphs, Ars Combinatorica. LXXVIII (2006) 297–308.
Z. Zhang and J.J. Yuan, Degree conditions for restricted-edge-connectivity and isoperimetric-edge-connectivity to be optimal, Discrete Math. 307 (2007) 293–298.
Refbacks
- There are currently no refbacks.
ISSN: 2338-2287
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.