and other short stories

姜子麟 Zilin Jiang

Arizona State University

June 21, 2024

Joint work with Ron Aharoni, Joseph Briggs and Ron Holzman

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.

Definition Given a family $\mathcal{F}$ of subsets $F_1, \dots, F_m$ of $E$ a subset $\{x_1, \dots, x_r\}$ is rainbow if there exist distinct $i_1, \dots, i_r$ such that $x_1 \in F_{i_1}, \dots, x_r \in F_{i_r}$

Recurring theme How large does $\mathcal{F}$ need to be to guarantee a rainbow set satisfying property $\mathcal{P}$ assuming that every member of $\mathcal{F}$ satisfies $\mathcal{P}$?

Recurring theme How large does $\mathcal{F}$ need to be to guarantee a rainbow set satisfying property $\mathcal{P}$ assuming that every member of $\mathcal{F}$ satisfies $\mathcal{P}$?

Drisko's theorem improved by Aharoni and Berger

$2n-1$ matchings of size $n$ in a bipartite graph have a rainbow matching of size $n$

Aharoni, Holzman and J.

$rn - r + 1$ fractional matchings of size $n$ in an $r$-partite hypergraph have a rainbow fractional matching of size $n$

Recurring theme How large does $\mathcal{F}$ need to be to guarantee a rainbow set satisfying property $\mathcal{P}$ assuming that every member of $\mathcal{F}$ satisfies $\mathcal{P}$?

Odd cycles: $\mathcal{P} =$ "being an odd cycle in a fixed $K_n$"

How many odd cycles in a fixed $K_n$ are needed to guarantee a rainbow odd cycle?

$n = 3$? Three odd cycles suffice

What's the answer in general?

How many odd cycles in a fixed $K_n$ are needed to guarantee a rainbow odd cycle?

Answer $n$ odd cycles suffice

Proof Take a maximal rainbow forest

At least one odd cycle is not used

This odd cycle is fully contained in a rainbow tree

Some edge doesn't respect bipartition of rainbow tree

Add that edge to the rainbow tree

How many odd cycles in a fixed $K_n$ are needed to guarantee a rainbow odd cycle?

Answer $n$ odd cycles suffice

Example $n-1$ identical cycles of length $n$ have no rainbow cycle

When $n$ is odd: $n$ odd cycles are necessary

When $n$ is even: $n-1$ odd cycles are necessary

Question Improvement for even $n$?

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

Think about $n = 5$...

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

Pruned cactus

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

# cycles $=$ # vertices $- 1$

No rainbow cycle

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.

In a pruned cactus, (a) each cycle of length $n$ repeats $n-1$ times, and (b) # cycles $=$ # vertices $- 1$

Pruned cactus of odd cycles has even number of cycles

Corollary When $n$ is even, $n-1$ odd cycles suffice

Aharoni, Briggs, Holzman, and J.

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: Break into 3 cases.

- There exists $\mathcal{K} \subsetneq \mathcal{O}$ such that $v(\cup \mathcal{K}) \le \lvert \mathcal{K} \rvert + 1$
- Every odd cycle in $\mathcal{O}$ is Hamiltonian
- 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

Case 1: There exists $\mathcal{K} \subsetneq \mathcal{O}$ such that $v(\cup \mathcal{K}) \le \lvert \mathcal{K} \rvert + 1$

Proof sketch "induction helps"

By induction, $\mathcal{K}$ is a pruned cactus

Without loss of generality, $\mathcal{K}$ consists of identical odd cycles $O$

Each odd cycle $O_i$ in $\mathcal{O}\setminus\mathcal{K}$ is partitioned by $V(O)$ into arcs, one of which is odd, say $P_i$

After contracting $V(O)$, odd arc $P_i$ becomes odd cycle

(a) rainbow cycle exists, or (b) new odd cycles form a pruned cactus

Case 2: Every odd cycle in $\mathcal{O}$ is Hamiltonian

Proof sketch

Let $S$ be a rainbow star of maximum size

As in Case 1, contract the leaves of $S$ and apply induction

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

Given a matroid with ground set $E$, for every subsets $E_1, \dots, E_m$ of $E$, a rainbow independent set of size $m$ exists if and only if $\mathrm{rank}(\bigcup_{i\in I} E_i) \ge |I|$ for every $I \subseteq [m]$

Corollary

For every connected subgraphs $E_1, \dots, E_m$ in $K_{m+1}$ a rainbow spanning tree exists if and only if $v(\bigcup_{i\in I} E_i) \ge |I| + 1$ for every $I \subseteq [m]$

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 $\mathcal{K}' \subseteq \mathcal{O}'$, $v(\cup \mathcal{K}') \ge \lvert \mathcal{K}' \rvert + 1$

Rainbow spanning tree exists

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 cactus 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 $\cup \mathcal{O}_2$

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.

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

Example there exist 6 squares on 6 vertices without rainbow even cycles

Glue a new copy at one vertex to the union of the previous ones

Györi, independently Frankl, Füredi and Simonyi, independently Goorevitch and Holzman

More than $n^2/8$ distinct triangles in $K_n$ have a rainbow triangle

Observation Take $n/4$ disjoint pairs of vertices, and connecting each pair to each of the remaining $n/2$ vertices with a triangle

How many cycles of length $a\mathbb{Z}+b$ are needed in $K_n$ to guarantee a rainbow cycle of length $a\mathbb{Z}+b$?

Theorem $2n-1$ matchings of size $n$ in a bipartite graph have a rainbow matching of size $n$

Conjecture $2n$ matchings of size $n$ ~~in a bipartite graph~~ have a rainbow matching of size $n$

姜子麟 Zilin Jiang

Arizona State University

zilinj@asu.edu

Arizona State University

zilinj@asu.edu