Path degeneracy and applications

In this work, we relate girth and path-degeneracy in classes with sub-exponential expansion, with explicit bounds for classes with polynomial expansion and proper minor-closed classes that are tight up to a constant factor (and tight up to second order terms if a classical conjecture on existence of $g$-cages is verified). As an application, we derive bounds on the generalized acyclic indices, on the generalized arboricities, and on the weak coloring numbers of high-girth graphs in such classes. Along the way, we prove a conjecture proposed in [T.~Bartnicki et al., Generalized arboricity of graphs with large girth, Discrete Mathematics 342 (2019), no.~5, 1343--1350.], which asserts that, for every integer $k$, there is an integer $g(p,k)$ such that every $K_k$ minor-free graph with girth at least $g(p,k)$ has $p$-arboricity at most $p+1$.