Testing C_k-Freeness in Bounded Admissibility Graphs

Abstract

We study C_k-freeness in sparse graphs from a property testing perspective, specifically for graph classes with bounded r-admissibility. Our work is motivated by the large gap between upper and lower bounds in this area: C_k-freeness is known to be testable in planar graphs [Czumaj and Sohler, 2019], but not in graphs with bounded arboricity for k > 3 [Talya Eden et al., 2024]. There are a large number of interesting graph classes that include planar graphs and have bounded arboricity (e.g. classes excluding a minor), calling for a more fine-grained approach to the question of testing C_k-freeness in sparse graph classes.
One such approach, inspired by the work of Nesetril and Ossona de Mendez [Nešetřil and {Ossona de Mendez}, 2012], is to consider the graph measure of r-admissibility, which naturally forms a hierarchy of graph families A₁ ⊃ A₂ ⊃ … ⊃ A_∞ where A_r contains all graph classes whose r-admissibility is bounded by some constant. The family A₁ contains classes with bounded arboricity, the class A_∞ contains classes like planar graphs, graphs of bounded degree, and minor-free graphs. Awofeso ηl [Awofeso et al., 2025] recently made progress in this direction. They showed that C₄- and C₅-freeness is testable in A₂. They further showed that C_k-freeness is not testable in A_{⌊k/2⌋ -1} and conjectured that C_k-freeness is testable in A_{⌊k/2⌋}. In this work, we prove this conjecture: C_k-freeness is indeed testable in graphs of bounded ⌊k/2⌋-admissibility.