Exact Turán numbers of two vertex-disjoint paths
Miao Dong, Bo Ning, Long-Tu Yuan, Xiao-Dong Zhang
TL;DR
The paper determines the exact Turán number ex(n, P_{k1} ∪ P_{k2}) for two vertex-disjoint odd paths with k1 ≥ k2 ≥ 3, establishing ex(n, P_{k1} ∪ P_{k2}) = max{ c(n, k1, k2), ex(n, P_{k1}), h(n, k1+k2-2, ⌊k1/2⌋+⌊k2/2⌋-1) }. This result completes the k=2 case of the Yuan–Zhang conjecture in a complete form when odd paths are considered, building on the classical ex(n, P_k) values and Erdős–Gallai results for matchings. The authors introduce three new path-tools—circumference/min-degree/clique inequalities, a stability version of Kopylov’s cycle theorem, and a refined Posá lemma—and define extremal templates H_k(s), H_k(M_2), and H_k(P_3) to manage small-n and 2-connected cases. The approach combines 2-connected reductions, refined rotation techniques, and stability arguments to derive tight upper bounds and characterize the extremal graphs, yielding a principled framework that advances the broader program of exact Turán numbers for disjoint subgraphs.
Abstract
The Turán number of a graph $H$ is the maximum number of edges in any graph of order $n$ that does not contain $H$ as a subgraph. In 1959, Erd\H os and Gallai obtained a sharp upper bound of Turán numbers for a path of arbitrary length. In 1975, Faudree and Schelp, and independently in 1977, Kopylov determined the exact values of Turán numbers of paths with arbitrary length. In this paper, we determine the Turán number of two vertex-disjoint paths of odd order at least 4. Together with previous works, we determine the exact Turán numbers of two vertex-disjoint paths completely. This confirms the first $k=2$ case of a conjecture proposed by Yuan and Zhang in 2021, which generalizes the Turán number formula of paths due to Faudree-Schelp, and Kopylov in a broader setting. Our main tools include a refinement of Pósa's rotation lemma, a stability result of Kopylov's theorem on cycles, and a recent inequality on circumference, minimum degree, and clique number of a 2-connected graph.