Rainbow odd cyclesand other short stories

Joint work with Ron Aharoni, Joseph Briggs and Ron Holzman

Romanian Master in Mathematics 2020

Let $n \ge 3$ be an integer. In a country there are $n$ airports and $n$ airlines operating two-way flights. For each airline, there is an odd integer $m \ge 3$, and $m$ distinct airports $c_1, \dots, c_m$ where the flights offered by the arline are exactly those between the following pairs of airports: $c_1$ and $c_2$, $c_2$ and $c_3$, ..., $c_{m-1}$ and $c_m$, $c_m$ and $c_1$. Prove that there is a closed route consisting of an odd number of flights where no two flights are operated by the same airline.

Romanian Master in Mathematics 2020

Let $C_1, C_2, \dots, C_n$ be subgraphs of a complete graph $K_n$. If $C_1, \dots, C_n$ are odd cycles, then there exists a rainbow odd cycle.

Definition Given a family (multiset) $\mathcal{F}$ of subsets of $E$, a subset $R \subseteq E$ with $\sigma \colon R \to \mathcal{F}$ is rainbow
if $e \in \sigma(e)$ for every $e \in R$ and $\sigma$ is injective

Problem Given a property $\mathcal{P}$, find smallest $m$ such that for every family $\mathcal{F}$ if $\lvert \mathcal{F} \rvert \ge m$ and every member of $\mathcal{F}$ satisfies $\mathcal{P}$, then there exists a rainbow set $R$ with $\mathcal{P}$.

Bárány's colorful Carathéodory theorem For every family of $n+1$ subsets of $\mathbb{R}^n$, each containing $\vec{a}$ in its convex hull, there exists a rainbow set with the same property.

Problem Given a property $\mathcal{P}$, find smallest $m$ such that for every family $\mathcal{F}$ if $\lvert \mathcal{F} \rvert \ge m$ and every member of $\mathcal{F}$ satisfies $\mathcal{P}$, then there exists a rainbow set $R$ with $\mathcal{P}$.

Theorem (Drisko '98, Aharoni and Berger '09) For every family of $2n-1$ matchings, each of size $n$, in a bipartite graph, there is a rainbow matching of size $n$.

Theorem (Aharoni, Kotlar and Ziv '18') For every $2n-2$ matchings, each of size $n$, in a bipartite graph, if there exists no rainbow matching of size $n$, then ...

Theorem (Aharoni, Kotlar and Ziv '18') For every $2n-2$ matchings, each of size $n$, in a bipartite graph, if there exists no rainbow matching of size $n$, then ...

$(n-1)$ red matchings + $(n-1)$ blue matchings

RMM 2020 For every family of $n$ odd cycles in $K_n$, there exists a rainbow odd cycle.

Proof Take a maximal rainbow forest
At least one odd cycle is not used
This odd cycle is fully contained in a rainbow tree
An edge doesn't respect bipartition of rainbow tree

RMM 2020 For every family of $n$ odd cycles in $K_n$, there exists a rainbow odd cycle.

Observation For $n-1$ identical cycles, each of length $n$, there exists no rainbow cycle.

RMM 2020 is sharp for odd $n$.

Question Improvement for even $n$?

For every family of $n-1$ odd cycles in $K_n$, if there exists no rainbow odd cycle, then ...

Definition A family $\mathcal{O}$ of cycles is a pruned cactus if all the cycles are identical to a cycle on $\lvert \mathcal{O} \rvert + 1$ vertices, or $\mathcal{O}$ can be partitioned into two pruned cacti $\mathcal{O}_1$ and $\mathcal{O}_2$ such that $\cup \mathcal{O}_1$ and $\cup \mathcal{O}_2$ share exactly one vertex

Each cycle of length $n$ repeats $n-1$ times

Observations

No rainbow cycle

Each odd cycle repeats even number of times

Theorem (Aharoni, Briggs, Holzman, and J.)

For every family of $n-1$ odd cycles in $K_n$, if there exists no rainbow odd cycle, then ...

the family is a pruned cactus.

Observation Every pruned cactus that consists of odd cycles only has even number of cycles

Corollary When $n$ is even, for every family of $n-1$ odd cycles in $K_n$, there exists a rainbow odd cycle.

Theorem For every family $\mathcal{O}$ of $n$ odd cycles in $K_{n+1}$, if no rainbow odd cycle, then $\mathcal{O}$ is a pruned cactus.

Proof sketch Break into 3 cases:

1. There exists $\mathcal{K} \subsetneq \mathcal{O}$ such that $v(\cup \mathcal{K}) \le \lvert \mathcal{K} \rvert + 1$
2. Every odd cycle in $\mathcal{O}$ is Hamiltonian
3. For every $\mathcal{K} \subsetneq \mathcal{O}$, $v(\cup \mathcal{K}) > \lvert \mathcal{K} \rvert + 1$, and some odd cycle in $\mathcal{O}$ is not Hamiltonian

Proof of Case 3 For every $\mathcal{K} \subsetneq \mathcal{O}$, $v(\cup \mathcal{K}) > \lvert \mathcal{K} \rvert + 1$, and some odd cycle in $\mathcal{O}$ is not Hamiltonian

Suppose $\mathcal{O} = \{O_1, \dots, O_n\}$ and $v \not\in O_n$

Consider $\mathcal{O}' = \{O_1-v, \dots, O_n - v\}$

They are connected subgraphs, and
for every $\mathcal{K}' \subseteq \mathcal{O}'$, $v(\cup \mathcal{K}') \ge \lvert \mathcal{K}' \rvert + 1$

Rado's matroid theorem: $\mathcal{O}'$ has a rainbow tree spanning $K_{n+1} - v$

Odd cycle $O_n$ is fully contained in this rainbow tree ...

Rainbow odd cycles

Proposition For every family of $n$ cycles in $K_n$, there exists a rainbow cycle

Observation For a pruned cacti with $n-1$ cycles (on $n$ vertices), there exists no rainbow cycle.

For every family of $n-1$ cycles in $K_n$, if there exists no rainbow cycle, then ...

Definition A family $\mathcal{O}$ of cycles is a saguaro if $\mathcal{O}$ is a pruned cactus, or $\mathcal{O}$ can be partitioned into two saguaros $\mathcal{O}_1$ and $\mathcal{O}_2$ and a single cycle $O$ such that $\cup \mathcal{O}_1$ and $\cup \mathcal{O}_2$ share no vertex, and $O$ is an even cycle that alternates between $\cup \mathcal{O}_1$ and $\cup \mathcal{O}_2$

Theorem (Aharoni, Briggs, Holzman, and J.)

For every family of $n-1$ cycles in $K_n$, if there exists no rainbow cycle, then ...

the family is a saguaro.

Rainbow even cycles

Theorem (Zichao Dong and Zijian Xu)
For every family of $\lfloor{(6n-1)/5}\rfloor$ even cycles in $K_n$, there exists a rainbow even cycle.

Open problems

Characterize families of $\lfloor{(6n-1)/5}\rfloor - 1$ even cycles in $K_n$ with no rainbow even cycle

Theorem (Drisko '98, Aharoni and Berger '09) For every $2n-1$ matchings, each of size $n$, in a bipartite graph, there exists a rainbow matching of size $n$.

Conjecture For every $2n$ matchings, each of size $n$, in a bipartite graph, there is a rainbow matching of size $n$.