2021/11/14 by Karim Chaira, Oleksiy Dovgoshey, Chaira, Karim +3 · 1 citation
Mathematics · #41A50 #Combinatorics (math.CO) #FOS: Mathematics #General Topology (math.GN) #Primary: 05C60. Secondary: 54E35 #math.CO #math.GN #msc:05C60. #msc:41A50 #msc:54E35
paper · pdf · doi:10.48550/arxiv.2111.07289
20 pages, 2 figures
arxiv created 2022/01/21 · arxiv updated 2022/01/24
We say that a bipartite graph G(A, B) with fixed parts A, B is proximinal if there is a semimetric space (X, d) such that A and B are disjoint proximinal subsets of X and all edges \a, b\ satisfy the equality d(a, b) = dist(A, B). It is proved that a bipartite graph G is not isomorphic to any proximinal graph iff G is finite and empty. It is also shown that the subgraph induced by all non-isolated vertices of a nonempty bipartite graph G is a disjoint union of complete bipartite graphs iff G is isomorphic to a nonempty proximinal graph for an ultrametric space.