Problem
What is the smallest order of a signed simple graph to which every planar signed simple graph admits a homomorphism?
The answer is known to be between 10 and 40. Moreover, the only possible candidate on 10 vertices is \( \Large A_9 \).
References
Alon-Marshall [1998]: Noga Alon, Timothy H. Marshall. Homomorphisms of edge-colored graphs and Coxeter groups. Journal of Algebraic Combinatorics 8 (1998), no. 1, 5-13. [DOI]
Naserasr-Rollová-Sopena [2015]: Reza Naserasr, Edita Rollová, Éric Sopena. Homomorphisms of signed graphs. Journal of Graph Theory 79 (2015), no. 3, 178-212. [DOI]
Ochem-Pinlou-Sen: Pascal Ochem, Alexandre Pinlou, Sagnik Sen. Homomorphisms of 2-edge-colored triangle-free planar graphs. Journal of Graph Theory 85 (2017), no. 1, 258-277. [DOI]