Zilin Jiang
Arizona State University
June 8, 2023
Joint work with Ron Aharoni, Joseph Briggs, and Ron Holzman
There are $n$ airports and $n$ airlines. Each airline operates cyclically between an odd number of airports. Prove that there is a closed route consisting of an odd number of flights where no two flights are operated by the same airline.
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
# cycles $=$ # vertices $- 1$
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:
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 $\mathcal{O}' = \{O_1-v, \dots, O_{n-1} - v\}$
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 ...
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.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.
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 $n$.