On the maximally edge-connected square graphs with degree sequence conditions

@misc{li2026maximally,
  title = {{On the maximally edge-connected square graphs with degree sequence conditions}},
  author = {Kai Li and Yingzhi Tian},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25386},
  eprint = {2610.25386},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25386},
}

TY  - UNPB
TI  - On the maximally edge-connected square graphs with degree sequence conditions
AU  - Kai Li
AU  - Yingzhi Tian
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25386
UR  - https://arxiv.org/abs/2610.25386
AB  - Let G=(V(G),E(G)) be a finite connected simple graph. The edge-connectivity λ(G) of G is the minimum number of edges that must be removed to make G disconnected. We call G maximally edge-connected when λ(G)=δ(G), where δ(G) is the minimum degree of G. The square graph G^2 of G is the graph with vertex set V(G), where two distinct vertices u and v are adjacent in G^2 precisely when their distance in G is at most two. In this paper, first we prove that if λ(G)≥4 and λ(G^2)<δ(G^2), then λ(G^2)>λ(G)^3/2−5/4λ(G), which extends the result in \cite{ChenXu} from the edge-connectivity range 4≤λ(G)≤7 to λ(G)≥4. Then we show that G^2 is maximally edge-connected if one of the following conditions hold: (i) δ(G)≥⌊(n+2)/4⌋, or δ(G)≤⌊(n+2)/4⌋−1 and ∑_i=1^2kd_n+1−i≥ k(n−4)−2δ(G)+3, where k is an integer satisfying 2≤ k≤δ(G)+2 and d_1≥ d_2≥⋯≥ d_n are the degrees of G; (ii) d_n−δ(G)+d_n≥(3n−9)/4.
ER  -