Notes on Aharoni’s rainbow cycle conjecture

Much of the research on the Caccetta-Häggkvist conjecture has focused on the directed triangle case ($r=\lceil\frac{n}{3}\rceil$). A natural strategy is to increase the outdegree condition until one can prove the existence of a directed triangle. The best result in this direction is the following result of Hladký et al. [12], which uses the flag algebra machinery developed by Razborov [19].

Similarly, for Aharoni’s conjecture, one can ask how large must the colour classes be to ensure a rainbow cycle of length at most $\lceil\frac{n}{r}\rceil$? The first non-trivial bound was proven by Hompe et al. [15].

Theorem 2.2 was subsequently improved by Hompe and Spirkl [17], who removed the $\log r$ term.

One appealing aspect of Aharoni’s conjecture is that we can ask what happens when the number of colour classes is different from the number of vertices. Note that this question does not even make sense in the digraph setting. To be precise, we now define a function $f(n,t,r)$ which will be useful to state many of the results that appear in this survey. The rainbow girth of an edge-coloured graph $G$, denoted $\operatorname{rg}(G)$, is the length of a shortest rainbow cycle in $G$. If $G$ does not contain a rainbow cycle, then $\operatorname{rg}(G)=\infty$. Let

where the maximum is taken over all simple edge-coloured graphs $G$ with $n$ vertices and at least $t$ colours, such that each colour class has size at least $r$.

We can rephrase Aharoni’s conjecture via our function $f(n,t,r)$ as follows.

We believe that there is no reason to restrict attention to the case $t=n$, and that the following question is of independent interest.

An important special case of Question 2.5 is when $r=1$, which was considered by Bollobás and Szemerédi [7].

In other words, Theorem 2.6 asserts that every $n$-vertex graph with at least $n+k$ edges contains a cycle of length at most $\frac{2(n+k)}{3k}(\log k+\log\log k+4)$. This is a key tool used in many of the results presented in this survey.

Hompe and Spirkl [17] also obtained the following bounds when the number of colours is more than the number of vertices.

When the number of colours is less than the number of vertices, DeVos et al. [9] obtained the following tight bounds for $r=2$.

In the proof of the Caccetta-Häggkvist conjecture (assuming Aharoni’s conjecture) given in the Introduction, the colour classes of the derived edge-colouring are all stars. Therefore, it is natural to ask what happens when the colour classes are not stars. Note that if a colour class is not a star, then it must contain a matching of size 2, or it is a triangle.

In the extreme case when all colour classes contain a matching of size 2, Aharoni and Guo [5] proved that there is a much shorter rainbow cycle than the $\lceil\frac{n}{2}\rceil$ bound which follows from [9].

Kevin Hendrey (private communication) proved that a $\mathcal{O}(\log n)$ bound also holds in the case when all colour classes are a triangle. The same proof also appears in a recent paper of Aharoni et al. [1].

Some mixed cases were investigated by Guo [10].

The rainbow triangle case ($r=\lceil n/3\rceil$) of Aharoni’s conjecture states that $f(n,n,\lceil n/3\rceil)\leqslant 3$. This is of course still open since the directed triangle case of the Caccetta-Häggkvist conjecture is still open. However, there have been partial results which increase the number of colours or sizes of the colour classes. Two such results were obtained by Aharoni et al. [4].

Both of these results have been subsequently improved by Hompe et al. [16].

Aharoni et al. [3] proved the following theorem showing when a simple edge-coloured graph with 3 colours contains a rainbow triangle. Their theorem actually implies Mantel’s theorem.

They also prove that the constant $\frac{26-2\sqrt{7}}{81}$ in Theorem 2.17 is best possible. However, their extremal example has rainbow 2-cycles. Our first new result is the following sharp thresholds for $f(n,3,r)$.

The Caccetta-Häggkvist conjecture is known to hold for small values of $r$. The $r=1$ case is trivial, the $r=2$ case was proven by Caccetta and Häggkvist [8]; the $r=3$ case was proven by Hamidoune [11]; and the $r\in\{4,5\}$ cases were proven by Hoàng and Reed [13]. For Aharoni’s conjecture, the $r=2$ case was proven by DeVos et al. [9].

In Section 3, we will prove the following result for $r=3$.

For the Caccetta-Häggkvist conjecture, it is natural to ask if there is a version which takes into account all the outdegrees rather than just the minimum outdegree. Seymour (see [14]) proposed the following generalization. Given a digraph $D$ with no sink⁴⁴4A sink is a vertex with outdegree zero., define

Note that in the case that all outdegrees are $r$, then $\psi(D)=\frac{n}{r}$, so Conjecture 2.21 implies the Caccetta-Häggkvist conjecture. Unfortunately, Conjecture 2.21 was disproved by Hompe [14]. However, Aharoni et al. [1] proved that "half" of Conjecture 2.21 holds.

Theorem 2.22 has a natural generalization in the rainbow setting. Given an edge-coloured graph $G$, define

where the sum is taken over all colour classes $A$ of $G$.

In the case that all colour classes have size at most 2, Aharoni et al. [1] proved the following strengthening of Conjecture 2.23.

Note that Theorem 2.24 is a strengthening of Theorem 2.19 since it allows colour classes of size 1.

We can generalise Aharoni’s conjecture to any setting in which the notion of ‘cycle’ makes sense. One natural candidate is that of a matroid. For the reader unfamiliar with matroids, we introduce all the necessary definitions now. For a more thorough introduction to matroids, we refer the reader to Oxley [18].

A matroid is a pair $M=(E,\mathcal{C})$ where $E$ is a finite set, called the ground set of $M$, and $\mathcal{C}$ is a collection of subsets of $E$, called circuits, satisfying

1. (1)

   $\emptyset\notin\mathcal{C}$,

2. (2)

   if $C^{\prime}$ is a proper subset of $C\in\mathcal{C}$, then $C^{\prime}\notin\mathcal{C}$,

3. (3)

   if $C_{1}$ and $C_{2}$ are distinct members of $\mathcal{C}$ and $e\in C_{1}\cap C_{2}$, then there exists $C_{3}\subseteq(C_{1}\cup C_{2})\setminus\{e\}$.

We now give some examples of matroids. Let $G$ be a graph. We will consider two different matroids with ground set $E(G)$. The circuits of the first matroid are the (edges of) cycles of $G$. This is the cycle matroid of $G$, denoted $M(G)$. A matroid is graphic if it is isomorphic to the cycle matroid of some graph. A cocycle of $G$ is an inclusion-wise minimal edge-cut of $G$. The collection of cocycles of $G$ is also a matroid, called the cocycle matroid of $G$, and is denoted $M(G)^{*}$. A matroid is cographic if it is isomorphic to the cocycle matroid of some graph.

Let $\mathbb{F}$ be a field. An $\mathbb{F}$-matrix is a matrix with entries in $\mathbb{F}$. Let $A$ be an $\mathbb{F}$-matrix whose columns are labelled by a finite set $E$. The column matroid of $A$, denoted $M[A]$, is the matroid with ground set $E$ whose circuits correspond to the minimal (under inclusion) linearly dependent columns of $A$. A matroid is representable over $\mathbb{F}$ if it is isomorphic to $M[A]$ for some $\mathbb{F}$-matrix $A$. A matroid is binary if it is representable over the two-element field, and it is regular if it is representable over every field.

In order to formulate Aharoni’s conjecture for matroids, we need to define simple matroids and how to express the number of vertices of a graph as a matroid parameter. We do this now. A matroid is simple it it does not contain any circuits of size $1$ or $2$. A set $I\subseteq E$ is independent if it does not contain a circuit. The rank of $X\subseteq E$ is the size of a largest independent set contained in $X$, and is denoted $r_{M}(X)$. The rank of $M$ is $r(M)\coloneqq r_{M}(E)$. Notice that the number of vertices of a connected graph $G$ is $r(M(G))-1$. Thus, Aharoni’s conjecture can be phrased in the language of matroids as follows.

One way to generalise Conjecture 1.1 is to replace ‘graphic matroid’ with some larger superclass of matroids. In [9], it was shown that one cannot replace ‘graphic matroid’ by ‘binary matroid’ in Conjecture 2.25.

The main result of [9] can be phrased in matroid language as follows.

In [9], it is also proved that the matroid analogue of Theorem 2.19 holds for cographic matroids.

Regular matroids are a well-studied superclass of graphic matroids, and are ‘essentially’ graphic or cographic via Seymour’s regular matroid decomposition theorem [20]. Therefore, by combining Theorem 2.27, Theorem 2.28, and Seymour’s regular matroid decomposition theorem, it may be possible to prove the following conjecture.

Of course, we can also consider Question 2.5 for various classes of matroids. Bérczi and Schwarcz [6] obtained one such result.

Bérczi and Schwarcz [6, Theorem 4] also show that it is possible to characterize binary matroids as exactly those matroids which do not admit a specific type of colouring with no rainbow circuits. They also prove that if a simple graph $G$ has an edge-colouring with no rainbow cycle, where each colour class has size at most 2, then $G$ is independent in the 2-dimensional rigidity matroid.⁶⁶6Equivalently, for every $X\subseteq V(G)$ with $|X|\geqslant 2$, $|E(G[X])|\leqslant 2|X|-3$.

We begin by establishing some basic properties about graphs which have at least $k$ more edges than vertices, which we call excess-$k$ graphs. The first property follows from  Theorem 2.6: for all $n\geqslant 4$ and $k\geqslant 2$, every $n$-vertex, excess-$k$ graph has a cycle of length at most $\frac{2(n+k)}{3k}(\log k+\log\log k+4)$. We require the following tighter bounds when $k\leqslant 2$.

The proof is easy and is omitted (see [9] for a proof of a stronger claim).

We now prove that the largest stable set of an excess-$k$ graph is roughly half the number of vertices.

We finish by establishing the following lemma about ‘minimal’ excess-$k$ rainbow subgraphs of an edge-coloured graph.

Let $G$ be a simple edge-coloured graph with $n$ vertices, $n$ colours, and each colour class of size exactly 3. An $r$-star is a star with $r$ edges. A colour class of $G$ is a star class if it is a 3-star. A vertex of $G$ is a star vertex if it is the centre of a star class, and is otherwise a non-star vertex. Let $S$ denote the set of star vertices of $G$, and $N$ denote the set of non-star vertices of $G$. Let $N^{\prime}$ be the set of non-star classes. Since star classes may be centred at the same vertex, we have $|N^{\prime}|\leqslant|N|$. We will do a case analysis depending on whether $|N|\geqslant 8$ or $|N|\leqslant 7$.

Throughout this section we suppose $|N|\geqslant 8$. Let $\{x,y\}\subseteq N$. We say that a colour class $A$ dominates $\{x,y\}$ if every edge in $A$ has an end in $\{x,y\}$.

We now choose $R$ to be an excess-2 rainbow subgraph of $G$ such that $V(R)$ is minimal under inclusion.

This completes the case $\lvert N\rvert\geqslant 8$, since we have found a rainbow cycle of length at most $\frac{2n}{5}+7$, which is better than the bound of $\frac{4n}{9}+7$ required by Theorem 3.1.

We complete the proof by considering the case $\lvert N\rvert\leqslant 7$.

Fix a non-star vertex $z\in V(G)$. Since $z$ is a non-star vertex, for every colour $a$, there is an edge $e_{a}$ coloured $a$ such that $e_{a}$ is not incident to $z$. Thus, there is a transversal $R$ of $G$ such that $z\notin V(R)$. In particular, $R$ is an excess-1 rainbow subgraph of $G$ such that $z\notin V(R)$. Let $R_{1}$ be an excess-1 rainbow subgraph of $G-z$ such that $V(R_{1})$ is minimal under inclusion.

Since $z\notin V(R_{2}^{\prime})$, $R_{2}^{\prime}$ is excess-1 (and rainbow). Let $R_{2}\subseteq R_{2}^{\prime}$ be excess-1 and rainbow such that $V(R_{2})$ is minimal under inclusion. In particular, $R_{2}$ has minimum degree at least 2.

Claim 3.12 completes the proof of Theorem 3.1.

This research was initiated at the Structural Graph Theory Downunder II workshop at the Mathematical Research Institute MATRIX (March 2022). We thank Ron Aharoni and Zach Hunter for carefully reading an earlier version of this paper and making some very helpful suggestions.