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