Some Partial Results on Linial's Conjecture for Matching-Spine Digraphs - Cruz, Jadder Bismarck de Sousa; Silva, Cândida Nunes da; Lee, Orlando (Anais do Encontro de Teoria da Computação (ETC), 2021)
Let $k$ be a positive integer. A partial $k$-coloring of a digraph $D$ is a set $\calC$ of $k$ disjoint stable sets and has weight defined as $\sum_{C \in \calC} |C|$. An optimal $k$-coloring is a $k$-coloring of maximum weight. A path partition of a digraph $D$ is a set $\calP$ of disjoint paths of $D$ that covers its vertex set and has $k$-norm defined as $\sum_{P \in \mathcal{P}} \min\{|P|,k\}$. A path partition $\calP$ is $k$-optimal if it has minimum $k$-norm. A digraph $D$ is matching-spine if its vertex set can be partitioned into sets $X$ and $Y$, such that $D[X]$ has a Hamilton path and the arc set of $D[Y]$ is a matching. Linial (1981) conjectured that the $k$-norm of a $k$-optimal path partition of a digraph is at most the weight of an optimal partial $k$-coloring. We present some partial results on this conjecture for matching-spine digraphs.