Diameter-two graphs whose mutual-visibility chromatic number equals their chromatic number

@misc{chaubal2026diametertwo,
  title = {{Diameter-two graphs whose mutual-visibility chromatic number equals their chromatic number}},
  author = {Siddhesh Chaubal},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.39430},
  eprint = {2610.39430},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.39430},
}

TY  - UNPB
TI  - Diameter-two graphs whose mutual-visibility chromatic number equals their chromatic number
AU  - Siddhesh Chaubal
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.39430
UR  - https://arxiv.org/abs/2610.39430
AB  - A set S of vertices of a graph G is a mutual-visibility set if every pair of vertices of S is joined by a shortest path whose internal vertices avoid S, and the mutual-visibility chromatic number χ_μ(G) is the least number of mutual-visibility sets into which V(G) can be partitioned. Klavžar, Kuziak, Valenzuela Tripodoro and Yero (2025) proved that χ_μ(G)≤χ(G) whenever G has diameter two, observed that equality holds for complete bipartite graphs, and asked whether there are any other diameter-two graphs attaining equality. We answer this question in the affirmative. We construct a graph G_11 of order 11 and diameter two with χ_μ(G_11)=χ(G_11)=3, and an infinite family of such graphs obtained from G_11 by replacing one vertex with pairwise non-adjacent copies of it.
ER  -