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List and total colorings of multiset permutation graphs

Italo J. Dejter

TL;DR

The article investigates multiset star transposition graphs $ST_k^\ell$ on all $\ell$-set permutations of length $k\ell$, introducing efficient$^\ell$-dominating sets (E$^\ell$-sets) and related partitions (SE$^\ell$-sets, $\Sigma$E-sets) to study colorings. It proves $ST_k^\ell$ is $(\ell-1)$-choosable for $k>1$, $\ell>2$, and admits total colorings; for $\ell=2$ it identifies an efficient coloring via $\Sigma_i^k$ and analyzes intricate 6-cycle color structures. The work extends to pancake permutation graphs $PC_k^\ell$, develops SE-/SigmaE-chains, and frames a cohesive structure generalizing Dejter–Serra results, while highlighting obstructions to segmental E-chains in certain variants. Overall, the paper advances combinatorial coloring theory for multiset permutation graphs, providing explicit constructions, partitions, and chain frameworks that unify list/coloring and total-coloring aspects with graph-theoretic architectures.

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.

List and total colorings of multiset permutation graphs

TL;DR

The article investigates multiset star transposition graphs on all -set permutations of length , introducing efficient-dominating sets (E-sets) and related partitions (SE-sets, E-sets) to study colorings. It proves is -choosable for , , and admits total colorings; for it identifies an efficient coloring via and analyzes intricate 6-cycle color structures. The work extends to pancake permutation graphs , develops SE-/SigmaE-chains, and frames a cohesive structure generalizing Dejter–Serra results, while highlighting obstructions to segmental E-chains in certain variants. Overall, the paper advances combinatorial coloring theory for multiset permutation graphs, providing explicit constructions, partitions, and chain frameworks that unify list/coloring and total-coloring aspects with graph-theoretic architectures.

Abstract

Let and be positive integers. The multiset star transposition graph ST has as vertices the -strings on symbols, each symbol repeated times, and edges given by the transpositions with (). It is shown for and that ST is -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 to that of an efficient dominating-set and applied to the graphs ST\,, showing they admit vertex partitions that generalize the Dejter-Serra partitions of ST into E-sets, but not efficiently in the sense that the distance of each E-set be 3. Efficiently in such sense however, and the related 2-set pancake permutation graph PC, among other intermediate permutation graphs, are shown to admit total colorings with colors that determine partitions into E-sets, each with distance 3. Furthermore, associated E-chains are examined.
Paper Structure (9 sections, 16 theorems, 13 equations, 4 figures)

This paper contains 9 sections, 16 theorems, 13 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Multiset star transposition graphs
  3. SE$\,^\ell$-sets of $\ell$-set star transposition graphs
  4. List coloring of $\ell$-set star transposition graphs
  5. Total coloring of $\ell$-set star transposition graphs
  6. Efficient coloring of 2-set star transposition graphs
  7. $\Sigma$E-sets of star 2-set transposition graphs
  8. SE- and $\Sigma$E-chains of 2-set star transposition graphs
  9. From 2-set star to pancake permutation graphs

Key Result

Theorem 2

Let $0<\ell ,k\in\mathbb{Z}$ and let $ST^\ell_k\ne ST^1_2$. For each $i\in[k]$, the multiset permutations $v_0\dots v_{k\ell -1}$ of the string $0^\ell 1^\ell 2^\ell \cdots(k-1)^\ell$ that have their first entry $v_0$ equal to $i$ constitute an SE$\,^\ell$-set $S_i^k$ of the graph $ST^\ell_k$. Each

Figures (4)

  • Figure 1: Colorings of $ST_3^1=ST_3$, $ST_2^2$ (twice) and the Desargues graph $ST_2^3$.
  • Figure 2: On 2-choosability of the Desargues graph $ST_2^3$.
  • Figure 3: Example for the proof of Theorem \ref{['teofig2']} and Example \ref{['exfig2']}.
  • Figure 4: Cutout of toroidal subgraph in Corollary \ref{['cutout']} in relation to Example \ref{['exfig2']}.

Theorems & Definitions (44)

  • Example 1
  • Theorem 2
  • proof
  • Corollary 3
  • proof
  • Corollary 4
  • proof
  • Corollary 5
  • proof
  • Example 6
  • ...and 34 more