Improved upper bounds on longest-path and maximal-subdivision transversals

Let G be a connected graph on n vertices. The Gallai number Gal(G) of is the size smallest set vertices that meets every maximum path in G. Grünbaum constructed with Gal(G)=3. Very recently, Long, Milans, and Munaro, proved Gal(G)≤8n3/4. This was first sub-linear upper bound terms n. We improve their to Gal(G)≤5n2/3. also tighten more general result Long et al. For multigraph M, we prove if L(M,G) M-subdivisions pairwise intersecting n≥m6, then has at most 5n2/3 Q∈L(M,G)