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Combinatorial spectra using polynomials

Sylwia Cichacz, Martin Dzúrik

Abstract

In this paper we would like to introduce some new methods for studying magic type-colorings of graphs or domination of graphs, based on combinatorial spectrum on polynomial rings. We hope that this concept will be potentially useful for the graph theorists.

Combinatorial spectra using polynomials

Abstract

In this paper we would like to introduce some new methods for studying magic type-colorings of graphs or domination of graphs, based on combinatorial spectrum on polynomial rings. We hope that this concept will be potentially useful for the graph theorists.
Paper Structure (8 sections, 18 theorems, 90 equations)

This paper contains 8 sections, 18 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Magic-type of labelings
  3. Anti-magic labeling
  4. Irregular labeling
  5. 1-2-3 conjecture
  6. Domination
  7. Domination number
  8. Edge Roman domination number

Key Result

Theorem 1.1

Let $p(x_1,\ldots,x_n)$ be a polynomial in $F[x_1,\ldots,x_n]$ of degree $t$, where $F$ is an arbitrary field. Suppose that the coefficient of the monomial of maximum degree in $\prod_{i=1}^nx^{t_i}_{{i}}$ is nonzero, where $t=\sum_it_i$ and each $t_i\geq 0$. Then, if $S_1,\ldots,S_n$ are subsets of

Theorems & Definitions (38)

  • Theorem 1.1: Combinatorial Nullstellensatz, Alon
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Lemma 2.3: Dz
  • Remark 2.4
  • Theorem 2.5
  • proof
  • Lemma 2.6
  • ...and 28 more