Total coloring and efficient domination applications to non-Cayley non-Schreier vertex-transitive graphs

TL;DR

This paper investigates total coloring and efficient dominating sets in specific non-Cayley, non-Schreier vertex-transitive graphs derived from transpositions, extending previous work on E-sets.

Contribution

It introduces total colorings with 2k-1 colors for certain transposition graphs and generalizes E-set partitions beyond prior Cayley graph results.

Findings

Total colorings achieved with 2k-1 colors

Existence of efficient dominating sets in these graphs

Partitioning of vertex sets into E-sets

Abstract

Let $0<k\in\mathbb{Z}$. Let the star 2-set transposition graph $ST^2_k$ be the $(2k-1)$-regular graph whose vertices are the $2k$-strings on $k$ symbols, each symbol repeated twice, with its edges given each by the transposition of the initial entry of one such $2k$-string with any entry that contains a different symbol than that of the initial entry. The pancake 2-set transposition graph $PC^2_k$ has the same vertex set of $ST^2_k$ and its edges involving each the maximal product of concentric disjoint transpositions in any prefix of an endvertex string, including the external transposition being that of an edge of $ST^2_k$. For $1<k\in\mathbb{Z}$, we show that $ST^2_k$ and $PC^2_k$, among other intermediate transposition graphs, have total colorings via $2k-1$ colors. They, in turn, yield efficient dominating sets, or E-sets, of the vertex sets of $ST^2_k$ and $PC^2_k$, and partitions into into $2k-1$ such E-sets, generalizing Dejter-Serra work on E-sets in such graphs.