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A sequel to the adventure of RGB-tilings to explore the Four Color Theorem

Shu-Chung Liu

Abstract

An approach of using RGB-tilings for proving the Four Color Theorem discussed in three previous work is expanded in this paper. A novel methodology and revisions for the methodology in the three aforementioned papers are discussed, and a previously derived result involving three degree-five vertices in a triangular graph is improved. Moreover, a treatment of a novel topic for a graph with six vertices of degree 5 in a dumbbell shape is presented.

A sequel to the adventure of RGB-tilings to explore the Four Color Theorem

Abstract

An approach of using RGB-tilings for proving the Four Color Theorem discussed in three previous work is expanded in this paper. A novel methodology and revisions for the methodology in the three aforementioned papers are discussed, and a previously derived result involving three degree-five vertices in a triangular graph is improved. Moreover, a treatment of a novel topic for a graph with six vertices of degree 5 in a dumbbell shape is presented.
Paper Structure (14 sections, 21 theorems, 4 equations, 25 figures, 2 tables)

This paper contains 14 sections, 21 theorems, 4 equations, 25 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Vertex-colorings and RGB-edge-colorings
  3. Tilings and 4-colorings for MPG's and semi-MPG's
  4. RGB-tilings of Types A, B, C and D on $EP-\{e\}$
  5. Synonym, equivalence, and congruence for $\mathcal{RGBT}(M)$
  6. Kempe chains around two adjacent vertices of degree 5 in $EP$
  7. Let $e$-diamonds and Kempe chains roll
  8. Symmetric RGB-tilings and $ATLAS$ for $T\!D:=(\{a,b\};\deg(a,b)=5)$
  9. Brief revision of Theorem \ref{['RGB3-thm:4deg5inDiamond']} and Lemma \ref{['RGB3-thm:5inTri565656']} in Liu2023III
  10. Four degree 5 vertices in a diamond
  11. Three degree 5 vertices in a triangle, continued
  12. Better result than Lemma \ref{['RGB4_nolineno:thm:5inTri565656']}
  13. Six vertices of degree 5 in a dumbbell
  14. Does $EP^\perp$ belong to $e\mathcal{MPGN}4$

Key Result

Theorem 1.4

Every R-tiling on a One Piece (which is either an MPG or an $n$-semi-MPG, $n\ge 3$) is grand.

Figures (25)

  • Figure 1: Corresponding RGB-edge-colorings
  • Figure 2: Red tile $v_2v_4$-diamond and a red half-tile
  • Figure 3: Three R-tilings; only the middle tiling induces a 4-coloring.
  • Figure 4: Two RGB-tilings; the right one induces a 4-coloring.
  • Figure 5: RGB-tilings of Types A, B, C, D on $EP-\{e\}$
  • ...and 20 more figures

Theorems & Definitions (37)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4: Theorem for One Piece, Theorem \ref{['RGB1-thm:RtilingOnePiece']} in Liu2023I
  • Theorem 1.5: The First Fundamental Theorem v1: for R-/RGB-tilings and 4-colorability, Theorem \ref{['RGB1-thm:4RGBtiling']} in Liu2023I
  • Corollary 1.6: Corollary \ref{['RGB1-thm:4RGBtiling2']} in Liu2023I
  • Theorem 1.7: Theorem \ref{['RGB1-thm:eMPG4']}(b) in Liu2023I
  • Lemma 1.8: Lemma \ref{['RGB1-thm:evenoddRGB']} in Liu2023I
  • Lemma 1.9
  • Theorem 1.10: Fundamental Theorem: necessary and sufficient conditions
  • ...and 27 more