Surprising identities for the greedy independent set on Cayley trees

Abstract

We prove a surprising symmetry between the law of the size $G_n$ of the greedy independent set on a uniform Cayley tree $ \mathcal{T}_n$ of size $n$ and that of its complement. We show that $G_n$ has the same law as the number of vertices at even height in $ \mathcal{T}_n$ rooted at a uniform vertex. This enables us to compute the exact law of the $G_n$. We also give a Markovian construction of the greedy independent set, which highlights the symmetry of $G_n$ and whose proof uses a new Markovian exploration of rooted Cayley trees which is of independent interest.

Figures (7)

• Figure 1: Example of the greedy independent set obtained on a tree of size $30$. The labels represent the order in which vertices are inspected in the construction of the greedy independent set. The green vertices are the active vertices whereas the red vertices are the blocked vertices.
• Figure 2: On the left, an illustration of the recursive equation in law for the number $H_n$ of vertices at even height on $\mathscr{T}_n$. In black at the bottom, the root vertex, in white its neighbours, and the next vertices to include are the black roots in each tree $T_i$. On the right, an illustration of the first step of the greedy algorithm. With our coupling with $\mathscr{T}_n$, the first vertex that we add in the greedy independent set is the root vertex of $\mathscr{T}_n$ (in green at the bottom and in red, its neighbours). The next vertices (in green) to inspect in each $T_i$ are then not necessarily the root vertices of $T_i$.
• Figure 3: On the left, a plane tree rooted at the black oriented edge. On the right, the usual representation of this plane tree rerooted at the green oriented edge.
• Figure 4: On the left, an illustration of the first step of the greedy algorithm for the independent set of edges. The first edge that we add in the independent set is the green root edge, and we block its neighbouring edges (red). The next edge to inspect is a uniform edge in each tree $T_i^{j}$, but can be taken as the root edge by invariance under rerooting. On the right, a plane tree $\mathbf{t}$ with black vertices and with edges colored as follows: we color the root edge in green, its neighbouring edges in red and reapply the procedure in each tree by taking first the root edge (in green). Its corresponding bi-type alternating plane tree $\mathbf{t}_g$ is obtained by considering the edges of $\mathbf{t}$ as the vertices of $\mathbf{t}_g$: the children of a green vertex in $\mathbf{t}_g$ correspond to its children followed by its brothers in $\mathbf{t}$, and a red vertex has a green child in $\mathbf{t}_g$ if the red corresponding edge has (at least) a child in $\mathbf{t}$.
• Figure 5: Example of a forest $\mathbf{F}_{10}^{\mathfrak{a}}$ which can be obtained after $10$ steps of exploration of a given tree $\mathbf{t}$ of size $18$. The edges which are still unknown are in dashed. If the vertex labeled $\mathfrak{a}( \mathbf{F}_{10}^{\mathfrak{a}}) = 16$ is the next peeled vertex, then the next edge to be added will be the dashed orange edge.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Theorem 3
• lemma 1
• Remark
• proof : Proof of the first half of Theorem \ref{['thm:height']}
• proof : Proof of the second half of Theorem \ref{['thm:height']}
• lemma 2
• proof : Proof of Theorem \ref{['thm:plane']}
• Definition 1