[Paper Review] An Anti-Ramsey Problem Concerning Complete Bipartite Graphs

Stephan Cho, Jay Cummings+2 more|arXiv (Cornell University)|Dec 9, 2014

Limits and Structures in Graph Theory2 references3 citations

TL;DR

This paper establishes a sufficient condition for any proper edge-coloring of a complete bipartite graph $K_{m,n}$ to contain a rainbow $K_{a,b}$ subgraph: if $m \geq a$ and $n > (a^2 - a + 1)(b - 1)$, then $K_{m,n} \rightarrow_R K_{a,b}$. The key contribution is proving this bound is tight for an infinite family of parameters using connections to finite projective planes, and introducing new anti-Ramsey numbers based on vertex and edge minimality in complete bipartite graphs.

ABSTRACT

We consider quadruples of positive integers $(a,b,m,n)$ with $a\leq b$ and $m\leq n$ such that any proper edge-coloring of the complete bipartite graph $K_{m,n}$ contains a rainbow $K_{a,b}$ subgraph. We show that any such quadruple with $a\leq m$ and $n>(a^2-a+1)(b-1)$ satisfies this property. We also show that the quadruple $(2,3,3,6)$ satisfies this property. We end with a conjecture.

Motivation & Objective

• To determine conditions under which every proper edge-coloring of a complete bipartite graph $K_{m,n}$ contains a rainbow $K_{a,b}$ subgraph.
• To investigate the minimal complete bipartite graphs $K_{m,n}$ that force the existence of a rainbow $K_{a,b}$ subgraph under proper edge-colorings.
• To define and compute new anti-Ramsey numbers based on vertex count ($AR_V$) and edge count ($AR_E$) of minimal such graphs.
• To explore the sharpness of the bound $n > (a^2 - a + 1)(b - 1)$ using finite projective planes when $a-1$ is a prime power.

Proposed method

• Use a canonical correspondence between properly edge-colored complete bipartite graphs and latin rectangles to translate the anti-Ramsey problem into a combinatorial problem on symbol distribution in rectangles.
• Apply induction on the number of columns to show that if $n > (a^2 - a + 1)(b - 1)$, then a rainbow $a \times b$ subrectangle must exist in the corresponding $a \times n$ latin rectangle.
• Leverage Singler’s theorem on cyclic automorphisms of projective planes to construct $a \times (a^2 - a + 1)$ latin rectangles without any rainbow $a \times 2$ subrectangles when $a-1$ is a prime power.
• Use the structure of projective planes to prove that the bound $n > (a^2 - a + 1)(b - 1)$ is sharp for an infinite family of parameters.
• Define vertex anti-Ramsey number $AR_V(K_{a,b})$ as the minimal number of vertices in $K_{m,n}$ such that $K_{m,n} \rightarrow_R K_{a,b}$, and similarly define edge anti-Ramsey number $AR_E(K_{a,b})$.
• Use combinatorial counting and extremal arguments to prove that $AR_V(K_{2,b}) = 3b$ and $AR_E(K_{a,b}) = a^2(a-1)(b-1) + ab$ under certain conditions on $a$ and $b$.

Experimental results

Research questions

• RQ1 Under what conditions on $m$, $n$, $a$, and $b$ does every proper edge-coloring of $K_{m,n}$ contain a rainbow $K_{a,b}$ subgraph?
• RQ2 Is the bound $n > (a^2 - a + 1)(b - 1)$ sharp for the existence of a rainbow $K_{a,b}$ subgraph in $K_{m,n}$?
• RQ3 Can the minimal complete bipartite graph $K_{m,n}$ that forces a rainbow $K_{a,b}$ subgraph be characterized in terms of vertex or edge count?
• RQ4 What is the value of the vertex anti-Ramsey number $AR_V(K_{2,b})$ for $b \geq 2$?
• RQ5 Under what conditions does $AR_E(K_{a,b}) = a^2(a-1)(b-1) + ab$ hold?

Key findings

• The paper proves that if $m \geq a$ and $n > (a^2 - a + 1)(b - 1)$, then every proper edge-coloring of $K_{m,n}$ contains a rainbow $K_{a,b}$ subgraph.
• The bound $n > (a^2 - a + 1)(b - 1)$ is sharp for an infinite family of parameters: when $a-1$ is a prime power, there exists a proper edge-coloring of $K_{a, (a^2 - a + 1)(b - 1)}$ that avoids any rainbow $K_{a,b}$ subgraph.
• The vertex anti-Ramsey number $AR_V(K_{2,b})$ is exactly $3b$ for all integers $b \geq 2$, meaning $K_{2,3b-2}$ does not guarantee a rainbow $K_{2,b}$, but $K_{2,3b-1}$ does.
• The edge anti-Ramsey number $AR_E(K_{a,b})$ equals $a^2(a-1)(b-1) + ab$ when $a-1$ is a prime power and $b \geq a(a-1)$, and this value is minimal for such graphs.
• The construction of $a \times (a^2 - a + 1)$ latin rectangles without rainbow $a \times 2$ subrectangles is possible if and only if $a-1$ is a prime power, linking the extremal problem to finite projective planes.
• The paper introduces and computes the first known values of vertex and edge anti-Ramsey numbers for complete bipartite graphs in the context of proper edge-colorings.