[Paper Review] Anti-Ramsey numbers for paths

Long‐Tu Yuan|arXiv (Cornell University)|Feb 1, 2021

Limits and Structures in Graph Theory16 references4 citations

TL;DR

This paper resolves a longstanding conjecture by Erdős, Simonovits, and Sós by exactly determining the anti-Ramsey number for paths. It proves that the maximum number of colors in an edge-colored complete graph $K_n$ without a rainbow path on $k$ vertices is given by a precise formula involving binomial coefficients and parity-dependent corrections, using stability methods from extremal graph theory to establish tight bounds for all $n \geq k \geq 5$. The result confirms the optimality of two known constructions and provides the first exact formula for anti-Ramsey numbers of paths.

ABSTRACT

We determine the anti-Ramsey numbers for paths. This confirms a conjecture posed by Erdős, Simonovits and Sós in 1970s.

Motivation & Objective

• To resolve a conjecture from the 1970s by Erdős, Simonovits, and Sós regarding the exact value of the anti-Ramsey number for paths.
• To determine the maximum number of colors in an edge-colored complete graph $K_n$ that avoids a rainbow copy of a path $P_k$.
• To establish tight extremal bounds using stability results from extremal graph theory, extending prior partial results to all $n \geq k \geq 5$.
• To confirm that the two known constructions—coloring a $K_{k-2}$ with distinct colors and the rest with one color, or using a structured graph $H(n,k-1,\ell-1)$—are indeed optimal.
• To provide a complete characterization of the extremal edge-colored graphs that avoid rainbow paths, using structural graph-theoretic arguments.

Proposed method

• Leverages recent stability theorems by Füredi, Kostochka, Luo, and Verstraëte on $P_k$-free and $\mathcal{C}_k$-free graphs, applicable even for small $n$.
• Applies the connected Turán number $\mathrm{ex}_{\text{con}}(n, P_k)$ to bound the number of edges in $P_k$-free connected graphs.
• Uses the extremal graph construction $H(n,k,a)$, which partitions vertices into sets $A$, $B$, and $C$ with specific edge sets, to model extremal configurations.
• Employs a contradiction argument based on the number of colors in a hypothetical extremal coloring: if $K_n$ has more than $\mathrm{ar}(n,k)$ colors, it must contain a rainbow $P_k$.
• Analyzes the structure of the representing graph $L_n$ of the edge-colored $K_n$, showing it must be a subgraph of $H(n,k-1,1)$ or $H(n,k-1,\ell-1)$ under the given constraints.
• Performs case analysis based on the parity of $k$, distinguishing between odd and even $k$, and uses degree and connectivity arguments to derive contradictions when assumptions violate the anti-Ramsey bound.

Experimental results

Research questions

• RQ1 What is the exact value of the anti-Ramsey number $\mathrm{AR}(n, P_k)$ for paths on $k$ vertices in $K_n$?
• RQ2 Are the two known constructions—coloring a $K_{k-2}$ with distinct colors and the rest with one color, or using the $H(n,k-1,\ell-1)$ graph—optimal for avoiding rainbow paths?
• RQ3 Can stability results from extremal graph theory be used to derive exact anti-Ramsey numbers for paths, even for small $n$?
• RQ4 How does the anti-Ramsey number for paths depend on the parity of $k$, and what explains the $\epsilon = 1$ or $2$ correction term in the formula?
• RQ5 Is it possible to construct a rainbow path on $k$ vertices in any edge-colored $K_n$ with more than $\mathrm{ar}(n,k)$ colors?

Key findings

• The paper establishes the exact anti-Ramsey number for paths: $\mathrm{AR}(n, P_k) = \max\left\{\binom{k-2}{2}+1, \binom{\ell-1}{2} + (\ell-1)(n - \ell + 1) + \epsilon\right\}$, where $\ell = \lfloor(k-1)/2\rfloor$ and $\epsilon = 1$ if $k$ is odd, $\epsilon = 2$ otherwise.
• For $n \geq k \geq 5$, the maximum number of colors in an edge-colored $K_n$ without a rainbow $P_k$ is achieved precisely by the two known extremal constructions.
• The proof confirms that the two constructions are optimal: one based on coloring a $K_{k-2}$ with distinct colors and the rest with a single color, and the other based on the graph $H(n,k-1,\ell-1)$.
• The stability results of Füredi, Kostochka, Luo, and Verstraëte are applied to graphs of arbitrary size, enabling the derivation of exact bounds even when $n$ is not large.
• A contradiction is derived when assuming a coloring with more than $\mathrm{ar}(n,k)$ colors exists without a rainbow $P_k$, by showing such a coloring would force the existence of a rainbow path.
• The analysis shows that any such extremal coloring must result in a representing graph $L_n$ that is a subgraph of $H(n,k-1,1)$ or $H(n,k-1,\ell-1)$, and that deviations lead to contradictions via degree or connectivity arguments.