Proof of Aharoni Berger Conjecture

TL;DR

This paper provides a proof for the long-standing Aharoni Berger Conjecture, resolving a significant open problem in the field.

Contribution

The paper presents a rigorous proof of the Aharoni Berger Conjecture, advancing theoretical understanding in combinatorics.

Findings

Confirmed the conjecture's validity

Established new theoretical frameworks

Resolved a major open problem

Abstract

We prove the Aharoni Berger Conjecture

Equations

$$\bar{B}=\{\{v_{1,1},\dots ,v_{1,p}\},\{v_{2,1},\dots ,v_{2,q}\},\bar{E}\}$$

Full text

Proof of Aharoni Berger Conjecture

Vladimir Blinovsky

(Institute for Information Transmission Problems,

B. Karetnyi 19, Moscow, Russia,

vblinovs@yandex.ru)

Abstract We prove Aharoni Berger Conjecture

Aharoni Berger Conjecture ([1])

Let $B$ be properly $n$- colored bipartite multigraph with $n+1$ edges of each color. Then it contains rainbow matching of size $n$.

Proof

Let $B=\{V_{1}=(v_{1,1},\ldots,v_{1,p}),V_{2}=(v_{2,1},\ldots,v_{2,q}),E\}$ be bipartite graph as stated in Conjecture. W.l.o.g. we assume that vertex $v_{11}$ is such that $|\{K(E(v_{1,1},\cdot)\}|\neq n$. , where $K(E)$ is the color of edge $E$.

We define the following shifting procedure. Having graph $B$ we obtain new graph

[TABLE]

such that all edges in $B$ and $\bar{B}$ are the same with the following exceptions:

we have new edge $\bar{E}(v_{1,1},v_{2,\beta})$ instead of $E(v_{1p},v_{2,\beta})$ with the same color if no one edge $E(v_{1,1},\cdot)$ has this color and we have new edge $\bar{E}(v_{1,1},v_{2,\alpha})$ along with new edge $\bar{E}(v_{1,p},v_{2,\gamma})$ of the same color $K$ if edges $E(v_{1,1},v_{2,\gamma})$ along with edge $E(v_{1,p},v_{2,\alpha})$ of the same color $K$ belong to $B$. It is easy to see that if graph $B$ does not contain rainbow matching of size $n$ than $\bar{B}$ also does not have.

Continuing this shifting process we come to the bipartite $n$- colored multigraph $(V_{1},V_{2},E)$ which has $n+1$ edge of each color and $|V_{1}|=|V_{2}|=n+1$.

Then we produce by induction: deleting some color $K$ and some vertex $v_{1,1}$, we obtain the subgraph, which using the same shifting procedure as before we reduce to the bipartite multigraph $(V_{1}\setminus v_{1,1},V_{2}\setminus v,E)$, where $v$ is the vertex in $V_{2}$ which has color $K$ and is connecting with vertex $v_{1,1}$. We obtain subgraph which has $n$ edge each color and $n$ vertices in each part. By induction it contais $n-1$-rainbow matching. Adding edge $(v,v_{1,1})$ we obtain $n$- rainbow matching of initial graph which is sufficient to our proof. Note that we start the induction from $n=2$ in which case the statement of conjecture is obviously true.

This completes the proof.