Let $k$ be a positive integer. A \emph{partial $k$-coloring} of a digraph $D$ is a set $\calC$ of $k$ disjoint stable sets and has \emph{weight} defined as $\sum_{C \in \calC} |C|$. An \emph{optimal} $k$-coloring is a $k$-coloring of maximum weight. A \emph{path partition} of a digraph $D$ is a set $\calP$ of disjoint paths of $D$ that covers its vertex set and has \emph{$k$-norm} defined as $\sum_{P \in \mathcal{P}} \min\{|P|,k\}$. A path partition $\calP$ is \emph{$k$-optimal} if it has minimum $k$-norm. A digraph $D$ is \emph{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.
A total coloring for a graph G is an assignment of colors to the edges and vertices of G such that any pair of adjacent or incident elements have different colors. The least number of colors for which G has a total coloring is denoted χ ′′ (G). It is known that splitcomparability graphs have χ ′′ (G) at most ∆(G) + 2. In this work we show that certain split-comparability graphs with odd maximum degree have χ ′′ (G) = ∆(G) + 1.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.