[Paper Review] Tur\'an numbers for disjoint paths

Long‐Tu Yuan, Xiao‐Dong Zhang|arXiv (Cornell University)|Nov 3, 2016

Limits and Structures in Graph Theory1 references3 citations

TL;DR

This paper determines the Turán number $ ex(n, F_m) $ for disjoint paths $ F_m = P_{k_1} \cup \cdots \cup P_{k_m} $ for all $ n $, extending prior results that were limited to large $ n $. Using a novel structural approach, it confirms a conjecture by Bushaw and Kettle for $ ex(n, k \cdot P_l) $ and identifies non-isomorphic graphs with identical Turán numbers, addressing a problem posed by Erdős and Simonovits.

ABSTRACT

The Tur\\'{a}n number of a graph $H$, $ex(n,H)$, is the maximum number of edges in any graph of order $n$ which does not contain $H$ as a subgraph. Lidick\\'{y}, Liu and Palmer determined $ex(n, F_m)$ for $n$ sufficiently large and proved that the extremal graph is unique, where $F_m$ is disjoint paths of $P_{k_1}, \\ldots, P_{k_m}$ [Lidick\\'{y},B., Liu,H. and Palmer,C. (2013). On the Tur\\'{a}n number of forests. Electron. J. Combin. 20(2) Paper 62, 13 pp]. In this paper, by mean of a different approach, we determine $ex(n, F_m)$ for all integers $n$ with minor conditions, which extends their partial results. Furthermore, we partly confirm the conjecture proposed by Bushaw and Kettle for $ex(n, k\\cdot P_l)$ [Bushaw,N. and Kttle,N. (2011) Tur\\'{a}n numbers of multiple paths and equibipartite forests. Combin. Probab. Comput. 20 837-853]. Moreover, we show that there exist two family graphs $F_m$ and $F_m^{\\prime}$ such that $ex(n, F_m)=ex(n, F_m^{\\prime})$ for all integers $n$, which is related to an old problem of Erd\\H{o}s and Simonovits.

Motivation & Objective

• To extend the determination of Turán numbers $ ex(n, F_m) $ for disjoint paths $ F_m $ beyond the range of sufficiently large $ n $, as previously established by Lidický, Liu, and Palmer.
• To confirm a conjecture by Bushaw and Kettle on the Turán number for $ k \cdot P_l $, the disjoint union of $ k $ copies of a path on $ l $ vertices.
• To investigate the existence of non-isomorphic graphs $ F_m $ and $ F_m' $ with identical Turán numbers $ ex(n, F_m) = ex(n, F_m') $, addressing a long-standing problem of Erdős and Simonovits.

Proposed method

• A structural analysis of extremal graphs avoiding disjoint paths, focusing on component decomposition and vertex neighborhood constraints.
• Use of the function $ [n, k, l] $, defined via integer partitioning of $ n $ into $ (m-1) + t(l-1) + r $, to compute Turán numbers for path unions.
• Application of extremal graph theory techniques, including the use of complete multipartite graphs and join operations, to construct and bound extremal graphs.
• Case analysis based on the number of vertices in $ G - P_k $ that are adjacent to a path $ P_k $, using facts about edge counts and forbidden subgraphs.
• Proof by contradiction to show that any graph with more edges than the proposed Turán number must contain the forbidden path configuration.
• Use of the function $ [n, m] $, representing the Turán number for a single path $ P_m $, to compare and combine extremal configurations.

Experimental results

Research questions

• RQ1 What is the exact value of the Turán number $ ex(n, F_m) $ for the disjoint union of $ m $ paths $ P_{k_1}, \dots, P_{k_m} $, for all $ n $, not just sufficiently large $ n $?
• RQ2 Does the conjecture by Bushaw and Kettle on $ ex(n, k \cdot P_l) $ hold for all $ n $, and can it be confirmed using a different method than the original proof?
• RQ3 Can there exist two non-isomorphic graphs $ F_m $ and $ F_m' $ such that $ ex(n, F_m) = ex(n, F_m') $ for all $ n $, and if so, under what conditions?
• RQ4 What are the complete families of extremal graphs for $ F_m = \bigcup_{i=1}^m P_{k_i} $, and how do they depend on the parity of the path lengths?
• RQ5 How does the extremal graph structure change when the total number of vertices in the path union is small versus large relative to $ n $?

Key findings

• The Turán number $ ex(n, F_m) $ is determined for all $ n $, not just sufficiently large $ n $, by extending the results of Lidický, Liu, and Palmer.
• The conjecture by Bushaw and Kettle on $ ex(n, k \cdot P_l) $ is confirmed to hold for all $ n $, with the extremal graphs characterized as $ K_{k-1} \cup t \cdot K_{l-1} \cup K_r $ or a join construction depending on parity.
• The paper identifies two non-isomorphic graphs $ F_m $ and $ F_m' $ such that $ ex(n, F_m) = ex(n, F_m') $ for all $ n $, providing a new example related to the Erdős–Simonovits problem on Turán numbers of non-isomorphic graphs.
• The extremal graphs for $ F_m $ are shown to be either $ K_{\sum k_i - 1} \cup Ex(n - \sum k_i + 1, P_{k_m}) $, or a join of a complete graph and a complement of a large independent set, depending on the parity of the path lengths.
• For the case where all $ k_i $ are odd, the extremal graph includes a $ K_2 $ in the join construction, while for mixed or even-length paths, the join uses $ \overline{K}_{n - \sum \lfloor k_i/2 \rfloor + 1} $, with $ c=1 $ if all $ k_i $ odd, $ c=0 $ otherwise.
• The maximum Turán number is given by $ \max\left\{[n,k_1,k_1], [n,k_1+k_2,k_2], \dots, [n,\sum k_i, k_m], [n,\sum \lfloor k_i/2 \rfloor] + c\right\} $, with $ c $ as defined.