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Quasiperiods of Magic Labeling Quasipolynomials

Margaret Bayer, Amanda Burcroff, Tyrrell B. McAllister, Leilani Pai

Abstract

A magic labeling of a graph is a labeling of the edges by nonnegative integers such that the label sum over the edges incident to every vertex is the same. This common label sum is known as the index. We count magic labelings by maximum edge label, rather than index, using an Ehrhart-theoretic approach. In contrast to Stanley's 1973 work showing that the function counting magic labelings with bounded index is a quasipolynomial with quasiperiod $2$, we show by construction that the minimum quasiperiod of the quasipolynomial counting magic labelings with bounded maximum label can be arbitrarily large, even for planar bipartite graphs. Unfortunately, this rules out a certain Ehrhart-theoretic approach to proving Hartsfield and Ringel's Antimagic Graph Conjecture. However, we show that this quasipolynomial is in fact a polynomial for any bipartite graph with matching preclusion number at most $1$, which includes any bipartite graph with a leaf.

Quasiperiods of Magic Labeling Quasipolynomials

Abstract

A magic labeling of a graph is a labeling of the edges by nonnegative integers such that the label sum over the edges incident to every vertex is the same. This common label sum is known as the index. We count magic labelings by maximum edge label, rather than index, using an Ehrhart-theoretic approach. In contrast to Stanley's 1973 work showing that the function counting magic labelings with bounded index is a quasipolynomial with quasiperiod , we show by construction that the minimum quasiperiod of the quasipolynomial counting magic labelings with bounded maximum label can be arbitrarily large, even for planar bipartite graphs. Unfortunately, this rules out a certain Ehrhart-theoretic approach to proving Hartsfield and Ringel's Antimagic Graph Conjecture. However, we show that this quasipolynomial is in fact a polynomial for any bipartite graph with matching preclusion number at most , which includes any bipartite graph with a leaf.
Paper Structure (8 sections, 14 theorems, 20 equations, 2 figures)

This paper contains 8 sections, 14 theorems, 20 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unbounded completely fundamental labelings
  4. The Ehrhart quasipolynomial of "1D43A"2099
  5. Computing the Ehrhart Quasipolynomial of "1D43A"2099
  6. Minimum quasiperiods and finite differences
  7. Proof that "1D439"2099 has minimum quasiperiod "1D45B
  8. Graphs with Small Magic-Labeling Quasiperiods

Key Result

Corollary 2.2

The denominator of a polytope $P$ is equal to the least common multiple of the final coordinates of the completely fundamental elements of $\Phi(P)$.

Figures (2)

  • Figure 1: The polytopes $P_G$ and $Q_G$ from \ref{['exmp: polytopes\n and cones']} are shown in the $xy$-plane in blue and red, respectively. A portion of the homogenized cone over each polytope is shown in the corresponding color.
  • Figure 2: The construction of the graph $G_n$.

Theorems & Definitions (32)

  • Definition 2.1
  • Corollary 2.2
  • Example 2.3
  • Lemma 2.4: Sta1973
  • Proposition 2.5: Sta1973
  • Theorem 2.6
  • Example 2.7
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • ...and 22 more