[Paper Review] Rainbow matchings in bipartite multigraphs

János Barát, András Gyárfás+1 more|Repository of the Academy's Library (Library of the Hungarian Academy of Sciences)|May 7, 2015

Limits and Structures in Graph Theory9 references4 citations

TL;DR

This paper establishes an upper bound on the minimum number of matchings of size $n$ needed in a bipartite multigraph to guarantee a rainbow matching of size $n-k$. Using a refined application of Woolbright's alternating path argument, it proves that $N(n,k) \leq \left\lfloor \frac{k+2}{k+1}n \right\rfloor - (k+1)$, which resolves the problem asymptotically and improves prior bounds for $k=1$, confirming a key step toward the Brualdi-Stein conjecture.

ABSTRACT

Suppose that $k$ is a non-negative integer and a bipartite multigraph $G$ is the union of $$N=\left\lfloor \frac{k+2}{k+1}n ight floor -(k+1)$$ matchings $M_1,\dots,M_N$, each of size $n$. We show that $G$ has a rainbow matching of size $n-k$, i.e. a matching of size $n-k$ with all edges coming from different $M_i$'s. Several choices of parameters relate to known results and conjectures.

Motivation & Objective

• To determine the minimal number $N(n,k)$ of matchings of size $n$ required in a bipartite multigraph to ensure a rainbow matching of size $n-k$.
• To unify and generalize prior results on rainbow matchings, including Theorem 1 ($N(n,0) = 2n-1$) and Conjecture 1 ($N(n,1) = n$).
• To provide a quantitative improvement over existing bounds for the size of rainbow matchings in $1$-factorizations and multigraphs.
• To extend Woolbright's $\sqrt{n}$-error result to general bipartite multigraphs using a novel matching construction technique.
• To resolve the extremal case for $k \geq \lfloor n/2 \rfloor$, showing $N(n,k) = n-k$ is tight.

Proposed method

• Define $N = \left\lfloor \frac{k+2}{k+1}n \right\rfloor - (k+1)$ as the candidate number of matchings to ensure a rainbow matching of size $n-k$.
• Assume by contradiction that the maximum rainbow matching $R_1$ has size $t \leq n-k-1$, and define $A_0 = A \setminus V(R_1)$, $B_0 = B \setminus V(R_1)$.
• Construct a sequence of matchings $F_j \subset M_j$ of size $j(n-t)$, ensuring $V(F_j) \cap B_0 = \emptyset$ via inductive construction.
• Use alternating paths to derive a contradiction: if $F_{i+1}$ intersects $B_0$, an alternating path can be formed that improves $R_1$, violating maximality.
• Derive the inequality $(N-t)(n-t) \leq t$, which leads to $N \leq \frac{n}{n-t} + t - 1$, and substitute $t \leq n-k-1$ to reach a contradiction with the definition of $N$.
• Leverage the structure of disjoint sets $A_i, B_i$ and the function $f(S)$ mapping vertices in $B$ to their neighbors in $A$ under $R_1$ to control vertex coverage.

Experimental results

Research questions

• RQ1 What is the minimal number $N(n,k)$ of matchings of size $n$ needed in a bipartite multigraph to guarantee a rainbow matching of size $n-k$?
• RQ2 Can the bound $N(n,k) \leq \left\lfloor \frac{k+2}{k+1}n \right\rfloor - (k+1)$ be proven for all $0 \leq k < n$?
• RQ3 Does this bound improve upon known results for $k=1$, such as the conjecture that $n$ matchings of size $n$ suffice for a rainbow matching of size $n-1$?
• RQ4 How does this result extend Woolbright’s $\sqrt{n}$-error bound for $K_{n,n}$-factorizations to general bipartite multigraphs?
• RQ5 Is the bound tight for $k \geq \lfloor n/2 \rfloor$, and does it imply $N(n,k) = n-k$ in this range?

Key findings

• The paper proves $N(n,k) \leq \left\lfloor \frac{k+2}{k+1}n \right\rfloor - (k+1)$, providing a unified upper bound for the number of matchings needed to ensure a rainbow matching of size $n-k$.
• For $k \geq \lfloor n/2 \rfloor$, the bound simplifies to $N(n,k) = n-k$, which is optimal and matches the trivial lower bound.
• When $k=0$, the bound reduces to $N(n,0) \leq 2n-1$, matching Theorem 1 and confirming its tightness.
• For $k=1$, the result implies that $\left\lfloor \frac{3}{2}n \right\rfloor - 2$ matchings of size $n$ suffice for a rainbow matching of size $n-1$, improving on prior bounds.
• The bound extends Woolbright’s $n - \sqrt{n}$ result to general bipartite multigraphs, showing that $k = \lfloor \sqrt{n} \rfloor$ yields $n - \sqrt{n}$ rainbow matching size.
• The proof technique, based on alternating paths and vertex coverage control, provides a general framework applicable beyond the $K_{n,n}$ case.