Spreads of degrees in maximal planar graphs -- an exposition

@misc{caro2026spreads,
  title = {{Spreads of degrees in maximal planar graphs -- an exposition}},
  author = {Yair Caro and Riste Škrekovski and Christina Zarb},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11395},
  eprint = {2610.11395},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11395},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Spreads of degrees in maximal planar graphs -- an exposition
AU  - Yair Caro
AU  - Riste Škrekovski
AU  - Christina Zarb
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11395
UR  - https://arxiv.org/abs/2610.11395
AB  - For a graph G and a set B⊆ V(G), the spread sp(B) of B is the difference between the largest and the smallest degree in G of a vertex of B, and for an integer k≥ 0 the parameter sp(G,k) is the largest cardinality of a set B with sp(B)≤ k. Caro, Lauri and Zarb asked for the minimum of sp(G,k) over the maximal planar graphs of order n; we write MP(n,δ,k) for this minimum over the maximal planar graphs of order n and minimum degree δ∈ {3,4,5}. This manuscript is intended as an exposition of the subject of spread in the degree sequence of a graph G, with emphasis on maximal planar graphs. We apply the general lower bound of the companion paper \cite{P1} to this class, together with further ideas, some based on Caro--West and Caro--Lauri--Zarb and some new. In particular we asymptotically determine the values of MP(n,δ,k) for every pair (δ,k) with δ∈ {3,4,5} and k≥ 0. This solution recovers the case MP(n,δ,0), which is the repetition number rep(G) introduced by Caro and West.
ER  -