List and total colorings of multiset permutation graphs

TL;DR

This paper investigates the colorability of multiset permutation graphs, introducing generalized domination concepts and demonstrating total colorings for specific classes of these graphs.

Contribution

It generalizes the notion of efficient domination sets to multiset permutation graphs and proves their total colorability under certain conditions.

Findings

Multiset star transposition graphs are $(\ell-1)$-choosable.

These graphs admit total colorings with $2k-1$ colors.

Vertex partitions into E-sets with specific distance properties are established.

Abstract

Let $k$ and $\ell$ be positive integers. The multiset star transposition graph ST$_k^\ell$ has as vertices the $k\ell$-strings $v_0\cdots v_{k\ell-1}$ on $k$ symbols, each symbol repeated $\ell$ times, and edges given by the transpositions $(v_0\;v_i)$ with $v_i\ne v_0$ ($0<i<k\ell$). It is shown for $k>1$ and $\ell>2$ that ST$_k^\ell$ is $(\ell-1)$-choosable and that, as a result, admits total colorings. In order to prove such assertions, the notion of efficient domination set (or E-set) of a graph is generalized for $\ell>1$ to that of an efficient dominating$\,^\ell$-set and applied to the graphs ST$_k^\ell$\,, showing they admit vertex partitions that generalize the Dejter-Serra partitions of ST$_k^1$ into E-sets, but not efficiently in the sense that the distance of each E$^\ell$-set be 3. Efficiently in such sense however, $ST^2_k$ and the related 2-set pancake permutation graph PC$^2_k$, among other intermediate permutation graphs, are shown to admit total colorings with $2k-1$ colors that determine partitions into $2k-1$ E-sets, each with distance 3. Furthermore, associated E-chains are examined.