[Paper Review] Bipartite Rainbow Numbers of Matchings

Xueliang Li, Jianhua Tu+1 more|ArXiv.org|Oct 30, 2006

Limits and Structures in Graph Theory5 references4 citations

TL;DR

This paper determines the exact bipartite rainbow number $ rb(K_{m,n}, kK_2) $ for all $ m \geq n \geq 1 $ and $ k \geq 1 $, establishing that the minimum number of colors guaranteeing a rainbow matching of size $ k $ in $ K_{m,n} $ is $ m(k-2) + 2 $ when $ m > k $, and $ k $ otherwise. The result extends Schiermeyer's earlier work on rainbow matchings in complete graphs to the bipartite setting using extremal graph theory and structural case analysis.

ABSTRACT

Given two graphs $G$ and $H$, let $f(G,H)$ denote the maximum number $c$ for which there is a way to color the edges of $G$ with $c$ colors such that every subgraph $H$ of $G$ has at least two edges of the same color. Equivalently, any edge-coloring of $G$ with at least $rb(G,H)=f(G,H)+1$ colors contains a rainbow copy of $H$, where a rainbow subgraph of an edge-colored graph is such that no two edges of it have the same color. The number $rb(G,H)$ is called the {\it rainbow number of $H$ with respect to $G$}, and simply called the {\it bipartite rainbow number of $H$} if $G$ is the complete bipartite graph $K_{m,n}$. Erdős, Simonovits and Sós showed that $rb(K_n,K_3)=n$. In 2004, Schiermeyer determined the rainbow numbers $rb(K_n,K_k)$ for all $n\geq k\geq 4$, and the rainbow numbers $rb(K_n,kK_2)$ for all $k\geq 2$ and $n\geq 3k+3$. In this paper we will determine the rainbow numbers $rb(K_{m,n},kK_2)$ for all $k\geq 1$.

Motivation & Objective

• To extend Schiermeyer's results on rainbow numbers for matchings in complete graphs to the bipartite setting.
• To determine the exact value of the bipartite rainbow number $ rb(K_{m,n}, kK_2) $ for all $ m \geq n \geq 1 $ and $ k \geq 1 $.
• To characterize extremal edge-colorings avoiding rainbow matchings of size $ k $ in $ K_{m,n} $.
• To establish a complete classification of rainbow numbers for matchings in complete bipartite graphs using extremal graph theory and structural case analysis.

Proposed method

• Uses extremal graph theory to determine $ ext(m,n,(k-1)K_2) $, the maximum number of edges in a bipartite graph without a $ (k-1)K_2 $ subgraph.
• Applies Hall's Marriage Theorem via Lemma 2.1 to bound the size of maximum matchings in bipartite graphs with given deficiency.
• Employs case analysis based on the structure of maximum matchings and neighborhood sets to bound edge counts in colorings avoiding rainbow $ kK_2 $.
• Analyzes edge colorings by fixing colors on key edges and using induction to force the existence of rainbow matchings in extremal configurations.
• Considers special graph structures such as $ SG_1 $ (a $ K_{m-1,m-1} $ with a pendant edge) and $ SG_2 $ (a $ K_{m-1,m-1} $ plus an isolated vertex) as extremal cases.
• Uses contradiction arguments: assuming no rainbow $ kK_2 $ exists leads to edge count bounds that force the existence of such a matching.

Experimental results

Research questions

• RQ1 What is the exact value of the bipartite rainbow number $ rb(K_{m,n}, kK_2) $ for all $ m \geq n \geq 1 $ and $ k \geq 1 $?
• RQ2 How does the rainbow number for matchings in $ K_{m,n} $ depend on the parameters $ m $, $ n $, and $ k $?
• RQ3 What are the extremal edge-colorings of $ K_{m,n} $ that avoid rainbow matchings of size $ k $?
• RQ4 Can the rainbow number for $ kK_2 $ in $ K_{m,n} $ be determined uniformly across all $ m \geq n $, or does it depend on relative size?

Key findings

• The bipartite rainbow number $ rb(K_{m,n}, kK_2) $ is exactly $ m(k-2) + 2 $ when $ m > k $.
• When $ m = k $, the rainbow number is $ k $, and this value is achieved only when $ m = n = k $.
• For $ m \geq 3 $ and $ n \geq 2 $, $ rb(K_{m,n}, 2K_2) = 2 $, while $ rb(K_{2,2}, 2K_2) = 3 $, showing a small exception at $ m=n=2 $.
• The extremal graph achieving $ ext(m,n,(k-1)K_2) $ is $ K_{m,k-1} $, and this structure is critical in bounding the rainbow number.
• Any edge-coloring of $ K_{m,n} $ with more than $ m(k-2) + 2 $ colors must contain a rainbow matching of size $ k $, and this bound is tight.
• The proof shows that all extremal colorings leading to $ m(k-2)+2 $ edges are isomorphic to $ SG_1 $ or $ SG_2 $, and in all such cases, a rainbow $ kK_2 $ is unavoidable.