Bertrand Guenin - The Cycling Conjecture

A family of sets over a finite groundset where no set is properly contained in another is a clutter. A cover of a clutter is a subset of the groundset that intersects every member of the clutter. The maximum number of pairwise disjoint sets in a clutter is the packing number and the cardinality of the smallest cover is the covering number. Clearly, the packing number is at most the covering number and a clutter packs if both numbers coincide. Not every clutter packs, consider for instance,

Examples of clutters that pack,

We will describe (4) in more details later. Examples (3) and (4) both involves a parity condition. For (3) all circuits have even length and for (4) all edge cuts have an even number of edges. The Cycling conjecture will provide sufficient conditions for a clutter, satisfying a particular parity condition, to pack.

The conjecture,

Consider a clutter \(\mathcal F\) with an element \(e\) of the groundset. The deletion minor \(\mathcal F\setminus e\) is defined as \(\{S\in\mathcal F: e\notin\mathcal F\}\), and the contraction minor \(\mathcal F/e\) is defined as the inclusion-wise minimal sets in \(\{S-\{e\}:S\in\mathcal F\}\). A minor of \(\mathcal F\) is a clutter obtained by a sequence of contractions and deletions.

A signed matroid is a pair \((M,\Sigma)\) where \(M\) is a binary matroid and \(\Sigma\subseteq EM\). A cycle of \(M\) is even (resp. odd) if \(|C\cap\Sigma|\) is even (resp. odd). Clearly, the set of inclusion-wise minimal odd cycles of \((M,\Sigma)\) is a clutter. Clutters that can be obtained by this construction are said to be binary.

A clutter \(\mathcal F\) is Eulerian if the parity of the cardinality of all inclusion-wise minimal covers is the same. At first glance this condition may appear non intuitive, so let us analyze what it says for a clutter \(\mathcal F\) of \(T\)-cuts in a graph \(G\) with terminals \(T\). The inclusion-wise minimal covers of \(\mathcal F\) are \(T\)-joins, and the parity of all \(T\)-joins is the same exactly when the symmetric difference of any two \(T\)-joins is a set with even cardinality. As the difference of two \(T\)-joins is an Eulerian subgraph, the condition translates to the requirement that \(G\) be bipartite.

The Cycling Conjecture

An Eulerian binary clutter packs if it does not contain any of the following clutters as a minor,

the Lines of the Fano,
the odd circuits of \(K_5\),
the complement of the cuts of \(K_5\), and
the \(T\)-joins of the Petersen graph where \(T\) is the set of all vertices.

For (1) the groundset of the clutter are the elements of the Fano matroids, and the members the lines. For (2) and (3) the groundset are the edges of the complete graph on \(5\) vertices. Note for (2) the members are the triangles and the pentagons and for (3) the members are triangles together with an independent edge and complete graphs on four vertices. For (4) the groundset are the edges of the Petersen graph and the member subgraphs where every vertex has odd degree. This conjecture, is hard as we will see that it implies the Four Colour Theorem.

A special case of the conjecture,

A signed-graph is a pair \((G,\Sigma)\) where \(G\) is a graph and \(\Sigma\) is a subset of the edges. We say that \(B\subseteq E(G)\) is even (resp. odd) if \(|B\cap\Sigma|\) is even (resp. odd). We call a subset of the edges of \((G,\Sigma)\) where \(s,t\in V(G)\) an odd \(st\)-walk if it is either an odd \(st\)-path, or it is the union of an even \(st\)-path \(P\) and an odd circuit \(C\) where \(P\) and \(C\) share at most one vertex. We consider the empty set to be an (even) \(st\)-path for the case where \(s=t\). Thus when \(s=t\), odd \(st\)-walks are odd circuits. We proved that the Cycling conjecture holds for the clutter of odd \(st\)-walks, more precisely

Theorem 1

An Eulerian clutter of odd \(st\)-walks packs if it does not contain any of the following clutters as a minor,

the Lines of the Fano,
the odd circuits of \(K_5\).

The proof and applications can be found in [1], see in [2] an extended abstract. A fractional version of this result can be found in [3].

Suppose \((G,\Sigma)\) is a signed-graph with vertices \(s,t\) and let \(\mathcal F\) be the clutter of odd \(st\)-walks of \((G,\Sigma)\). Then \(\mathcal F\) is Eulerian if either \(s=t\) and the degree of every vertex is even, or \(s\neq t\) and \(\deg(s)-|\Sigma|\) and the degree of every vertex in \(V(G)-\{s,t\}\) are even. The theorem states that in that case either: the maximum number of pairwise (edge) disjoint odd \(st\)-walks is equal to the minimum number of edges needed to intersect all odd \(st\)-walk, or \(\mathcal F\) must have as a minor the clutter of the lines of the Fano or the clutter of odd circuits of \(K_5\).

When \(s=t\), the condition that \({\mathcal F}\) is Eulerian says that all vertices of \(G\) have even degree. Moreover, in that case odd \(st\)-walks are odd circuits. Hence, Theorem 1 becomes,

Theorem 2

Let \(G\) be a graph that does not contain \(K_5\) as an odd minor and where every vertex has even degree. Then the minimum number of edges needed to intersect all odd circuits is equal to the maximum number of pairwise disjoint odd circuits.

A blocking vertex (resp. blocking pair) is a vertex (resp. pair of vertices) that intersects every odd circuit. Next we indicate a number of classes of signed-graphs for which the clutter of odd \(st\)-walks does not have as a minor the lines of the Fano or the clutter of odd circuits of \(K_5\).

Thus for each of the classes of signed-graphs (A)-(F), as long as the Eulerian condition holds, the minimum number of edges needed to intersect all odd \(st\)-walk is equal to the maximum number of disjoint odd \(st\)-walks.

Corollary

Let \((H,T)\) be a graft with \(|T|\leq 4\). Suppose that every vertex of \(H\) not in \(T\) has even degree and that all the vertices in \(T\) have degrees of the same parity. Then the maximum number of pairwise disjoint \(T\)-joins is equal to the minimum size of a \(T\)-cut.

Applying Theorem 1 to class (B) we can derive the two-commodity flow of Hu, Rothschild and Whinston, namely,

Corollary

Let \(H\) be a graph with vertices \(s_1,s_2,t_1,t_2\) where \(s_1\neq t_1\), \(s_2\neq t_2\), all of \(s_1,t_1,s_2,t_2\) have the same parity, and all the other vertices have even degree. Then the maximum number of pairwise disjoint paths that are between \(s_i\) and \(t_i\) for some \(i=1,2\), is equal to the minimum size of an edge subset whose deletion removes all \(s_1t_1\)- and \(s_2t_2\)-paths.

(\(\star\)) start from a plane graph with exactly two faces of odd length and distinct vertices \(s\) and \(t\), and identify \(s\) and \(t\).

Corollary

Let \(H\) be a graph as in (\(\star\)) and suppose that the length of the shortest odd circuit is \(k\). Then there exists cuts \(B_1,\ldots,B_k\) such that every edge \(e\) is in at least \(k-1\) of \(B_1,\ldots,B_k.\)

Suppose that \(H\) is as in (\(\star\)) and is loopless. Then by Corollary 6, there exists cuts \(\delta(U_1), \delta(U_2)\) such that every edge is in \(\delta(U_1)\cup\delta(U_2)\). It follows that for all \(i,j\in[4]\) where \(i\neq j\), \(U_i\cap U_j\) is a stable set. Hence, \(H\) is \(4\)-colourable.

The following conjecture would generalize the previous result as well as the 4-colour theorem,

Conjecture

Let \(H\) be a graph that does not contain \(K_5\) as an odd minor and suppose that the length of the shortest odd circuit is \(k\). Then there exists cuts \(B_1,\ldots,B_k\) such that every edge \(e\) is in at least \(k-1\) of \(B_1,\ldots,B_k\).

It can be shown that this conjecture is another special case of the Cycling Conjecture. It is open for planar graphs.