Coupon-Coloring and total domination in Hamiltonian planar triangulations

TL;DR

This paper investigates coupon-coloring and total domination in Hamiltonian planar triangulations, establishing bounds on the total domatic number and advancing understanding of coloring properties in these graphs.

Contribution

It determines the total domatic number for maximal outerplanar graphs and proves that Hamiltonian maximal planar graphs have a domatic number of at least two, addressing a conjecture.

Findings

Total domatic number in maximal outerplanar graphs determined

Hamiltonian maximal planar graphs have domatic number ≥ 2

Partially confirms a conjecture of Goddard and Henning

Abstract

We consider the so-called coupon-coloring of the vertices of a graph where every color appears in every open neighborhood, and our aim is to determine the maximal number of colors in such colorings. In other words, every color class must be a total dominating set in the graph and we study the total domatic number of the graph. We determine this parameter in every maximal outerplanar graph, and show that every Hamiltonian maximal planar graph has domatic number at least two, partially answering a conjecture of Goddard and Henning.