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A lower bound on forcing numbers based on height functions

Fateh Aliyev, Nikita Gladkov

Abstract

We establish a lower bound on the forcing numbers of domino tilings computable in polynomial time based on height functions. This lower bound is sharp for a 2n by 2n square as well as other cases.

A lower bound on forcing numbers based on height functions

Abstract

We establish a lower bound on the forcing numbers of domino tilings computable in polynomial time based on height functions. This lower bound is sharp for a 2n by 2n square as well as other cases.

Paper Structure

This paper contains 4 sections, 13 theorems, 20 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Characterizing the difference
  3. Minimum maximum excess
  4. Extension to the triangular lattice

Key Result

Theorem 1.1

The forcing number of a region $R$ on the square lattice is bounded below by

Figures (4)

  • Figure 1: Minimal forcing set in a $6 \times 6$ square.
  • Figure 2: Triangular lattice
  • Figure 3: Minimal and maximal tilings of a hexagon with sides $3$, $4$, $6$ with lozenges.
  • Figure 4: Forcing set for a hexagon with sides $3$, $4$, $6$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: Fou
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Theorem 3.1: Rid
  • Theorem 3.2: Lp
  • ...and 10 more