Tuza’s conjecture for binary geometries

A simple binary matroid is a pair $M=(E,P)$, where $P=\mathrm{PG}(n-1,2)$ is a finite-dimensional binary projective geometry of dimension $n-1$ and $E\subseteq P$. The rank of $M$ is 1 plus the dimension of the largest subgeometry of $P$ that contains $E$.

For matroids $M=(E_{1},P_{1})$ and $N=(E_{2},P_{2})$ we say that $M$ contains $N$ as a restriction if there exists a linear injection $\varphi\colon P_{2}\rightarrow P_{1}$ such that $\varphi(E_{2})\subseteq E_{1}$. We say that $M$ is $N$-free is $M$ does not contain $N$ as a restriction. We say that $M$ and $N$ are isomorphic if there exists a linear bijection $\varphi\colon P_{1}\rightarrow P_{2}$ such that $\varphi(E_{1})=E_{2}$.

This definition of simple binary matroid is essentially the same as the standard definition of such matroids, except that our matroids are equipped with an extrinsic ambient space.

We abuse notation and write $\mathrm{PG}(n-1,2)$ for both the $(n-1)$-dimensional binary projective geometry and the corresponding rank-$n$ simple binary matroid $(E,P)$ with $E=P=\mathrm{PG}(n-1,2)$. We refer to the matroids $\mathrm{PG}(1,2)$ and $\mathrm{PG}(2,2)=:F_{7}$ as the triangle and the Fano plane, respectively.

Let $G$ be a graph, let $A$ be its vertex-edge incidence matrix, and write $n$ for the rank of $A$. Let $A^{\prime}$ be obtained from $A$ by restriction to a subset of its rows that is a basis of its row space $\mathcal{R}(A)$.

The points of $P=\mathrm{PG}(n-1,2)$ can be identified with the nonzero binary vectors in $\mathbb{F}_{2}^{n}$. Let $E$ be the subset of $\mathrm{PG}(n-1,2)$ formed by the columns of $A^{\prime}$, then $(E,P)$ is a simple binary matroid. Although in our formalism $M(G)$ depends on the choice of $A^{\prime}$, all such choices yields isomorphic matroids, and we will write $M(G)$ for the resulting corresponding matroid. More generally, a matroid is called graphic if it can be obtained from a graph in this way.

In a similar fashion, if $A^{\prime\prime}$ is a matrix whose rows form a basis of the orthogonal complement $\mathcal{R}(A)^{\perp}$, we can use the columns of $A^{\prime\prime}$ to define a matroid $M^{*}(G)$ (as in the case of $M(G)$, this matroid is unique up to isomorphism), and we call a matroid cographic if it can be obtained from a graph in this way.

Triangles in $M(G)$ correspond to triangles in $G$, while triangles in $M^{*}(G)$ correspond to minimal edge cuts (bonds) of cardinality 3 in $G$ (which we will also call triads).

Let $M=(E,P)$ be a simple binary matroid. A (triangle) packing of $M$ is a collection of disjoint triangles contained in $M$; a (triangle) hitting set of $M$ is a subset $X\subseteq E$ such that $(E\setminus X,P)$ is triangle-free. We write $\nu(M)$ for the maximum size of a triangle packing in $M$, and $\tau(M)$ for the minimum size of a triangle hitting set.

Alternatively, the parameters $\nu(M)$ and $\tau(M)$ can be formulated as the objective value of integer programmes. Write $\mathcal{T}(M)$ for the collection of triangles of $M$, then

and

It is easily verified that $\nu(M(G))=\nu(G)$ and $\nu(M(G))=\tau(G)$ for any graph $G$.

We generalise the notion of weighted graphs, as discussed by Chapuy et al. [CDM⁺14], to matroids. Let $M=(E,P)$ be a simple binary matroid, and let $w\colon E\rightarrow\mathbb{Z}_{\geq 0}$ be a weight function; we refer to the pair $(M,w)$ as a weighted binary matroid. The parameters $\nu(M)$ and $\tau(M)$ are easily generalised to the weighted setting:

and

If $w_{e}=1$ for all $e\in E$, then $\nu_{w}(M)=\nu(M)$ and $\tau_{w}(M)=\tau(M)$. The weighted versions of these parameters allow us to talk about binary matroids that may contain non-trivial parallel classes. When $M=(E,P)$ and $w\colon E\rightarrow\mathbb{Z}_{\geq 0}$ is a weight function, we can define a related weight function $w^{\prime}\colon P\rightarrow\mathbb{Z}_{\geq 0}$ by setting $w^{\prime}(e)=w(e)$ if $e\in E$ and $w^{\prime}(e)=0$ otherwise. In that case, $\tau_{w^{\prime}}(P)=\tau_{w}(M)$ and $\nu_{w^{\prime}}(P)=\nu_{w}(M)$; thus, we may always assume that $E=P$.

Let $M=(E,P)$ be a simple binary matroid. It will be useful to encode the triangles of $M$ as a hypergraph $\mathbb{T}(M)$ on vertices $E$, in which a 3-set $T\in\binom{E}{3}$ forms a hyperedge if and only if $T$ is a triangle of $M$. The hypergraph $\mathbb{T}(M)$ is clearly 3-uniform, and as two triangles in $M$ intersect in at most one point, the hypergraph is linear as well. In terms of $\mathbb{T}(M)$, $\nu(M)$ is the size of a maximum matching in $\mathbb{T}(M)$, while $\tau(M)$ is the size of a minimum cover in $\mathbb{T}(M)$.

We will need the following standard result on hypergraphs.

A crown of size $k$ is a linear cycle on vertices $\{e_{1},\dots,e_{k},f_{1},\dots,f_{k}\}$ with edges $\{e_{i},f_{i},e_{i+1}\}$, $i\in[k]$ (where we identify $k+1$ with $1$), with the additional property that these are the only edges in which the $e_{i}$ are contained. Crowns were introduced as an inductive tool in [Tuz90, Lemma 2].

We prove the following reformulation of Theorem 4. We remark that Claims 7.1 and 7.2 are just Properties (a)–(c) in [CDM⁺14, Section 3] adapted to our context, and follow from the same argument.

A note on language: In the proof of the following lemma, we consider both a simple cographic matroid $M$ and a graph $G$ such that $M=M^{*}(G)$. We use the terms “triangle of $M$”, “triad in $G$”, and “hyperedge of $\mathbb{T}(M)$” interchangeably, depending on the context.

Consider $\mathrm{PG}(n-1,2)$. When $n=1$ or $n=2$, Tuza’s conjecture holds trivially, but when $n=3$ it fails. We will assume that $n\geq 4$.

The removal of a hyperplane makes $\mathrm{PG}(n-1,2)$ triangle-free, and no smaller set has the same property. It follows that

A spread is a partition of a projective geometry into lower-dimensional subgeometries. The following result, phrased here in matroidal terms, can be found in [Dem68, p. 29].

An immediate consequence of this result is that binary projective geometries of even rank can be partitioned into triangles, and hence

Binary projective geometries of odd rank cannot be partitioned into triangles; however partial spreads were studied by Beutelspacher [Beu75, Theorems 4.1–4.2], who showed that binary projective geometries of odd rank can be partitioned into triangles and four additional points (in fact, a 4-circuit, but that is not important here), and hence

Combining (1)–(3), Tuza’s conjecture for projective geometries of rank at least 4 follows.

For $n\geq k$, let $P=\mathrm{PG}(n-1,2)$ and let $Q$ be a subgeometry of $P$ of rank $n-k$. We refer to the matroid $(P-Q,P)$ as the Bose–Burton geometry $\mathrm{BB}(n,k,2)$. It is the maximum-size simple binary matroid of rank $n$ that does not contain $\mathrm{PG}(k-1,2)$ as a submatroid [BB66].

It is easily checked that $\mathrm{BB}(n,0,2)$ is empty and $\mathrm{BB}(n,1,2)$ is triangle-free. In both cases, Tuza’s conjecture holds trivially. For the remainder of this section, we will therefore assume that $k\geq 2$.

We next show that $\mathrm{BB}(n,2,2)$ can be partitioned into triangles, using a variant of the proof of [Beu75, Theorem 4.2].

Lemma 10 and Lemma 12 now readily imply Tuza’s conjecture for Bose–Burton geometries.

The critical number $\chi(M)$ of an $n$-dimensional simple binary matroid $M=(E,P)$ is the smallest integer $k\geq 0$ such that $M$ is a restriction of the Bose–Burton geometry $\mathrm{BB}(n,k,2)$; alternatively, it is the smallest integer $k\geq 0$ such that $P\setminus E$ contains a subgeometry of $P$ of dimension $n-k$. In particular, if $\chi(M)\leq 2$, then there is a rank-$(n-2)$ subgeometry of $P$ such that $E\subseteq P\setminus Q$. Matroids with critical number at most 2 do not have restrictions isomorphic to the Fano plane.