[Paper Review] Extremal graphs for wheels

Long‐Tu Yuan|arXiv (Cornell University)|Jan 7, 2020

Limits and Structures in Graph Theory18 references4 citations

TL;DR

This paper determines the exact Turán number for odd wheels $W_{2k+1}$ for sufficiently large $n$, showing that the extremal graphs are formed by a specific degree-optimized partition of vertices. It further confirms that the maximum number of edges not in any monochromatic copy of $W_{2k+1}$ in a 2-edge-coloring of $K_n$ equals the Turán number, thus verifying a conjecture by Keevash and Sudakov for odd wheels.

ABSTRACT

For a graph $H$, the Turán number of $H$, denoted by ex$(n,H)$, is the maximum number of edges of an $n$-vertex $H$-free graph. Let $g(n,H)$ denote the maximum number of edges not contained in any monochromatic copy of $H$ in a $2$-edge-coloring of $K_n$. A wheel $W_m$ is a graph formed by connecting a single vertex to all vertices of a cycle of length $m-1$. The Turán number of $W_{2k}$ was determined by Simonovits in the 1960s. In this paper, we determine ex$(n,W_{2k+1})$ when $n$ is sufficiently large. We also show that, for sufficiently large $n$, $g(n,W_{2k+1})=\mbox{ex}(n,W_{2k+1})$ which confirms a conjecture posed by Keevash and Sudakov for odd wheels.

Motivation & Objective

• To determine the exact Turán number $\mathrm{ex}(n, W_{2k+1})$ for odd wheels when $n$ is sufficiently large.
• To resolve a conjecture by Keevash and Sudakov stating that $g(n, H) = \mathrm{ex}(n, H)$ for $H = W_{2k+1}$, where $g(n, H)$ is the maximum number of edges not in any monochromatic copy of $H$ in a 2-edge-coloring of $K_n$.
• To characterize the structure of extremal graphs for odd wheels, showing they are not covered by classical extremal graph theory results due to the decomposition family not containing a linear forest.
• To establish that the extremal colorings for $W_{2k+1}$ are precisely those where the red graph is an extremal $W_{2k+1}$-free graph and the blue graph is its complement.

Proposed method

• Define $f(n,k) = \max\left\{n_0n_1 + \left\lfloor \frac{(k-1)n_0}{2} \right\rfloor + 1 : n_0 + n_1 = n \right\}$ as the candidate extremal edge count for $W_{2k+1}$.
• Use progressive induction on the number of vertices to analyze extremal 2-edge-colorings of $K_n$ avoiding monochromatic $W_{2k+1}$, focusing on the structure of the NIM-edges (non-monochromatic edges).
• Apply a vertex partitioning strategy dividing $V(K_n)$ into two large parts $B_1, B_2$ of size $n/2 + O(\sqrt{n})$, each inducing a red complete bipartite graph, and a small exceptional set $D$ of size $O(1)$.
• Establish bounds on the number of blue edges incident to vertices in $B_i \cup C_i$, showing that each such vertex has at most $k-1$ blue neighbors within the set, ensuring no blue $W_{2k+1}$ is formed.
• Use Lemma 5.3 to find a blue copy of $T(2N,2)$ in the coloring, and apply progressive induction to show that the extremal coloring must be of the form where the red graph is extremal for $W_{2k+1}$.
• Leverage the fact that all edges within $V_i = B_i \cup C_i \cup D_i$ are blue and have bounded degree ($\leq k-1$) to prove that the extremal graph is unique up to isomorphism.

Experimental results

Research questions

• RQ1 What is the exact value of the Turán number $\mathrm{ex}(n, W_{2k+1})$ for sufficiently large $n$?
• RQ2 Does $g(n, W_{2k+1}) = \mathrm{ex}(n, W_{2k+1})$ hold for all sufficiently large $n$, confirming the Keevash-Sudakov conjecture for odd wheels?
• RQ3 What structural properties do extremal graphs for $W_{2k+1}$ possess, especially given that the decomposition family of $W_{2k+1}$ does not contain a linear forest?
• RQ4 How can one characterize the extremal 2-edge-colorings of $K_n$ that avoid monochromatic copies of $W_{2k+1}$?
• RQ5 What is the role of the exceptional set $D$ in the extremal coloring, and how does its size affect the extremal edge count?

Key findings

• For $k \geq 3$, $\mathrm{ex}(n, W_{2k+1}) = f(n,k) = \max\left\{n_0n_1 + \left\lfloor \frac{(k-1)n_0}{2} \right\rfloor + 1 : n_0 + n_1 = n \right\}$ when $n$ is sufficiently large.
• For $k = 2$, $\mathrm{ex}(n, W_5) = \left(\lceil \frac{n}{2} \rceil + 1 \right) \left\lfloor \frac{n}{2} \right\rfloor$, which corresponds to the complete split graph $K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor} + e$.
• $g(n, W_{2k+1}) = \mathrm{ex}(n, W_{2k+1})$ for all sufficiently large $n$, confirming the Keevash-Sudakov conjecture for odd wheels.
• The extremal graphs for $W_{2k+1}$ are characterized as those where the vertex set is partitioned into two parts $V_1, V_2$ with $|V_1| = n_0$, $|V_2| = n_1$, $n_0 + n_1 = n$, and the edge set consists of all edges between $V_1$ and $V_2$, plus $\left\lfloor \frac{(k-1)n_0}{2} \right\rfloor + 1$ edges within $V_1$, maximizing the total edge count under $W_{2k+1}$-freeness.
• The extremal 2-edge-colorings of $K_n$ avoiding monochromatic $W_{2k+1}$ are precisely those where the red graph is an extremal $W_{2k+1}$-free graph and the blue graph is its complement, with the red graph being the unique extremal graph for $W_{2k+1}$.
• The size of the exceptional set $D$ in the progressive induction is bounded by a constant $N_1$, and all vertices in $B_i \cup C_i$ have at most $k-1$ blue neighbors within their own set, ensuring no blue $W_{2k+1}$ is formed.