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Mrs. Correct and Majority Colorings

Marcin Anholcer, Bartłomiej Bosek, Jarosław Grytczuk, Grzegorz Gutowski, Jakub Przybyło, Mariusz Zając

Abstract

A majority coloring of a directed graph is a vertex coloring in which each vertex has the same color as at most half of its out-neighbors. In this note we simplify some proof techniques and generalize previously known results on various generalizations of majority coloring. In particular, our unified and simplified approach works for paintability - an on-line analog of the list coloring.

Mrs. Correct and Majority Colorings

Abstract

A majority coloring of a directed graph is a vertex coloring in which each vertex has the same color as at most half of its out-neighbors. In this note we simplify some proof techniques and generalize previously known results on various generalizations of majority coloring. In particular, our unified and simplified approach works for paintability - an on-line analog of the list coloring.
Paper Structure (13 sections, 13 theorems, 10 equations)

This paper contains 13 sections, 13 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Coloring with tolerance
  3. List coloring
  4. Paintability
  5. Color ranks
  6. Paintability with ranks
  7. Undirected graphs
  8. The main results
  9. Results and proofs
  10. Undirected graphs
  11. Strongly connected directed graphs
  12. Directed graphs
  13. Discussion

Key Result

Theorem 1

Every positively-edge-weighted undirected graph is ranked-majority $1$-paintable.

Theorems & Definitions (13)

  • Theorem 1: Undirected, Ranked Paintability
  • Theorem 1: Directed, Ranked Paintability
  • Lemma 1
  • Theorem 2: Undirected, Ranked Paintability
  • Corollary 3: Undirected, Ranked Choosability
  • Corollary 4: Undirected, Non-uniform Paintability
  • Corollary 5: Undirected Paintability
  • Lemma 6
  • Lemma 7: Rank Reduction
  • Theorem 8: Directed, Ranked Paintability
  • ...and 3 more