Path degeneracy and applications
Y. Lin, P. Ossona de Mendez
TL;DR
This work analyzes how large girth in sparse graph classes enforces path-degeneracy, linking girth to structural sparsity across sub-exponential, polynomial, and minor-closed expansions. It provides explicit girth thresholds, notably via Lambert $W_{-1}$-based bounds, and shows these thresholds are tight up to constants; it also derives consequences for generalized arboricity, generalized acyclic chromatic indices, and weak coloring numbers, proving several conjectures for minor-closed classes. The methods combine shallow-minor density controls with path-reduction techniques to obtain linear growth rates in key parameters for high-girth graphs. The results have broad implications for structural graph theory and algorithmic applications on sparse graph classes, including tight bounds in $K_k$-minor-free settings and planarity-related instances.
Abstract
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 $p+1$.