[Paper Review] Extremal graph for intersecting odd cycles

Xinmin Hou, Yu Qiu+1 more|arXiv (Cornell University)|Oct 28, 2015

Limits and Structures in Graph Theory5 references3 citations

TL;DR

This paper determines the extremal graphs for intersecting odd cycles, proving that for sufficiently large n, the maximum number of edges in a graph without k intersecting q-cycles (q odd, q ≥ 5) is achieved uniquely by the Turán graph T_{n,2} with a complete bipartite graph K_{k-1,k-1} embedded in one partite set. The extremal edge count is e(T_{n,2}) + (k-1)^2, generalizing prior results on intersecting triangles and cliques.

ABSTRACT

An extremal graph for a graph $H$ on $n$ vertices is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $T_{n,r}$ be the Turán graph, which is the complete $r$-partite graph on $n$ vertices with part sizes that differ by at most one. The well-known Turán Theorem states that $T_{n,r}$ is the only extremal graph for complete graph $K_{r+1}$. Erdös et al. (1995) determined the extremal graphs for intersecting triangles and Chen et al. (2003) determined the maximum number of edges of the extremal graphs for intersecting cliques. In this paper, we determine the extremal graphs for intersecting odd cycles.

Motivation & Objective

• To determine the extremal graphs that maximize the number of edges without containing k intersecting odd cycles of length q ≥ 5.
• To generalize prior results on extremal graphs for intersecting triangles and cliques to the case of intersecting odd cycles.
• To establish the exact extremal edge count and characterize the unique extremal graph structure for large n.
• To prove that the extremal graph is uniquely the Turán graph T_{n,2} with a K_{k-1,k-1} embedded in one partite set.

Proposed method

• Uses the Turán graph T_{n,2} as the base structure, which is the complete balanced 2-partite graph on n vertices.
• Introduces a family of graphs F_{n,k} consisting of T_{n,2} with a K_{k-1,k-1} embedded in one partite set.
• Applies extremal graph theory techniques, including degree and matching number constraints via Lemma 2.1 and stability results from Lemma 2.2.
• Employs structural analysis and contradiction arguments to show that any extremal graph must have one partite set containing a K_{k-1,k-1} and the other with no internal edges.
• Uses degree sum and matching number bounds to restrict possible configurations and prove uniqueness.
• Applies Lemma 2.4 and Observation 2.3 to bound the number of edges in subgraphs with maximum degree ≤ 2 and control matching numbers.

Experimental results

Research questions

• RQ1 What is the maximum number of edges in a graph on n vertices that avoids k intersecting odd cycles of length q ≥ 5?
• RQ2 What is the unique extremal graph structure that achieves this maximum edge count for large n?
• RQ3 How does the extremal graph for intersecting odd cycles generalize previous results on intersecting triangles and cliques?
• RQ4 Under what conditions does the extremal graph consist of a Turán graph T_{n,2} with a complete bipartite subgraph embedded in one partite set?
• RQ5 Why is the edge count exactly e(T_{n,2}) + (k-1)^2, and why is this structure the only extremal one?

Key findings

• For all n ≥ n₁(k,q), the extremal number of edges ex(n, C_{k,q}) equals e(T_{n,2}) + (k-1)^2.
• The only extremal graphs for C_{k,q} are those in the family F_{n,k}, consisting of T_{n,2} with a K_{k-1,k-1} embedded in one partite set.
• The extremal graph is unique up to isomorphism for sufficiently large n.
• The proof establishes that any extremal graph must have one partite set containing a complete balanced bipartite graph K_{k-1,k-1} and the other with no internal edges.
• The edge count (k-1)^2 in the embedded K_{k-1,k-1} is tight and cannot be exceeded without creating a copy of C_{k,q}.
• The result generalizes Theorem 1.1 of Erdös et al. (1995) on k-fans to the case of intersecting odd cycles of odd length q ≥ 5.