Saturated Domino Coverings

TL;DR

This paper introduces the concept of saturated domino coverings and minimal fragment tilings, establishing their relationship with domination numbers and providing new sequences related to board coverings.

Contribution

It defines minimal fragment tilings and their connection to saturated domino coverings, linking these to domination numbers and introducing new OEIS sequences for board coverings.

Findings

Minimal fragment tilings correspond to maximal saturated domino coverings.

The size of minimal fragment tilings equals the domination number of the board.

New sequences counting saturated domino coverings are identified in OEIS.

Abstract

A domino covering of a board is saturated if no domino is redundant. We introduce the concept of a fragment tiling and show that a minimal fragment tiling always corresponds to a maximal saturated domino covering. The size of a minimal fragment tiling is the domination number of the board. We define a class of regular boards and show that for these boards the domination number gives the size of a minimal X-pentomino covering. Natural sequences that count maximal saturated domino coverings of square and rectangular boards are obtained. These include the new sequences A193764, A193765, A193766, A193767, and A193768 of OEIS.