List and total colorings of multiset permutation graphs

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.

💡 Research Summary

The paper introduces a new family of graphs called multiset star‑transposition graphs, denoted STₖ^ℓ, where k and ℓ are positive integers. The vertex set consists of all length‑kℓ strings over the alphabet {0,…,k‑1} in which each symbol appears exactly ℓ times (ℓ‑set permutations). Two vertices are adjacent if one can be obtained from the other by swapping the first entry with any other entry that contains a different symbol; this operation is called a star transposition. When ℓ=1 the graph reduces to the well‑known Cayley graph of the symmetric group Symₖ generated by the transpositions (0 i), i∈