Linial-Berge path partition duality

Definitions and mathematical statement

Let $D=(V,A)$ be a finite loopless digraph, allowing both opposite arcs between two vertices. A directed path is a sequence of distinct vertices with an arc from each vertex to its successor. A singleton vertex is also a path. A path partition $\mathcal P$ is a family of vertex-disjoint directed paths whose vertex sets partition $V$.

For an integer $k\ge1$, define the $k$-norm of a partition and its minimum by \[ \|\mathcal P\|_k=\sum_{P\in\mathcal P}\min\{|V(P)|,k\}, \qquad \pi_k(D)=\min_{\mathcal P}\|\mathcal P\|_k. \] An independent set in $D$ is a vertex set with no arc between any two of its distinct vertices, in either direction. Let $\alpha_k(D)$ be the maximum size of a union of $k$ independent sets. Equivalently, it is the largest number of vertices inducing a graph whose underlying undirected graph has a proper coloring with at most $k$ colors. Empty color classes are permitted.

The conjecture is $\pi_k(D)\le\alpha_k(D)$ for every finite $D$ and every positive integer $k$. Thus some partition should have total truncated path length at most the largest $k$-colorable vertex set. The independent sets do not merely induce acyclic digraphs: arcs inside one color class are prohibited altogether. Path lengths in the formula count vertices, not arcs, so singleton paths contribute one. Both optimization quantities are finite and attained.

Short English statement

Can every digraph be partitioned into directed paths whose total k-truncated vertex counts are bounded by its largest union of k independent sets?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3190 (OPG-170), OpenGarden. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Linial-Berge path partition duality. Original problem and discussion. http://www.openproblemgarden.org/op/linial_berge_path_partition_duality.