Distance labellings of Cayley graphs of semigroups

TL;DR

This paper explores how the structure of semigroups influences the minimum spans in their Cayley graph labelings, revealing structural characterizations and bounds for these spans.

Contribution

It characterizes semigroups based on minimum span restrictions and describes the structure of Cayley graphs with specific span properties, linking algebraic and graph-theoretic features.

Findings

Restrictions on minimum spans characterize combinatorial semigroups.

Certain restrictions are equivalent to semigroups being right zero bands.

The upper bound for minimum spans is sharp even for combinatorial semigroups.

Abstract

This paper establishes connections between the structure of a semigroup and the minimum spans of distance labellings of its Cayley graphs. We show that certain general restrictions on the minimum spans are equivalent to the semigroup being combinatorial, and that other restrictions are equivalent to the semigroup being a right zero band. We obtain a description of the structure of all semigroups $S$ and their subsets $C$ such that $\Cay(S,C)$ is a disjoint union of complete graphs, and show that this description is also equivalent to several restrictions on the minimum span of $\Cay(S,C)$. We then describe all graphs with minimum spans satisfying the same restrictions, and give examples to show that a fairly straightforward upper bound for the minimum spans of the underlying undirected graphs of Cayley graphs turns out to be sharp even for the class of combinatorial semigroups.