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From semi-total to equitable total colorings

I. J. Dejter

TL;DR

This work develops a constructive program to transform semi-total colorings into equitable total colorings for cubic graphs, leveraging a Kempe-like algorithm built around color-alternating paths (MCAPs). It introduces and exploits β- and γ-reductions, along with covering-graph lifting, to move from lacunar or non-equitable colorings to equitable total colorings across symmetric cubic graphs and cage graphs. Through extensive, concrete examples on the 3-cube, Möbius ladders, Desargues, Foster, McGee, Petersen, and other cages, the paper demonstrates how to obtain equitable total colorings and even 1-total-perfect lacunar STCs in various families. The results provide a versatile framework linking STCs, TCs, covering maps, and perfect codes to achieve equitable colorings in a broad class of cubic graphs.

Abstract

Independently posed by Behzad and Vizing, the Total Coloring Conjecture asserts that the total chromatic number of a simple connected graph $G$ is either $Δ(G)+1$ or $Δ(G)+2$, where $Δ(G)$ is the largest degree of any vertex of $G$. To decide whether a cubic graph $G$ has total chromatic number $Δ(G)+1$, even for bipartite cubic graphs, is NP-hard. The resulting problems and research persist even for total colorings that are equitable, namely with the cardinalities of the color classes differing at most by 1. Williams and Holroyd gave a new condition to solve total coloring problems via the introduction of semi-total colorings. We focus on how to obtain equitable total colorings of symmetric cubic graphs and cage graphs by means of a variation of Kempe'a 1879 graph-coloring algorithm. Such variation takes semi-total colorings to equitable ones.

From semi-total to equitable total colorings

TL;DR

This work develops a constructive program to transform semi-total colorings into equitable total colorings for cubic graphs, leveraging a Kempe-like algorithm built around color-alternating paths (MCAPs). It introduces and exploits β- and γ-reductions, along with covering-graph lifting, to move from lacunar or non-equitable colorings to equitable total colorings across symmetric cubic graphs and cage graphs. Through extensive, concrete examples on the 3-cube, Möbius ladders, Desargues, Foster, McGee, Petersen, and other cages, the paper demonstrates how to obtain equitable total colorings and even 1-total-perfect lacunar STCs in various families. The results provide a versatile framework linking STCs, TCs, covering maps, and perfect codes to achieve equitable colorings in a broad class of cubic graphs.

Abstract

Independently posed by Behzad and Vizing, the Total Coloring Conjecture asserts that the total chromatic number of a simple connected graph is either or , where is the largest degree of any vertex of . To decide whether a cubic graph has total chromatic number , even for bipartite cubic graphs, is NP-hard. The resulting problems and research persist even for total colorings that are equitable, namely with the cardinalities of the color classes differing at most by 1. Williams and Holroyd gave a new condition to solve total coloring problems via the introduction of semi-total colorings. We focus on how to obtain equitable total colorings of symmetric cubic graphs and cage graphs by means of a variation of Kempe'a 1879 graph-coloring algorithm. Such variation takes semi-total colorings to equitable ones.

Paper Structure

This paper contains 13 sections, 2 theorems, 7 equations, 20 figures.

Table of Contents

  1. Introduction
  2. Total colorings
  3. Semi-total colorings
  4. Equitable total colorings
  5. Beta and Gamma reductions
  6. TCs via covering graph maps
  7. Perfect and total perfect codes
  8. Symmetric cubic graphs
  9. Total reductions to equitable TCs
  10. Examples of TCs via covering graph maps
  11. Gamma-reductions to equitable STCs
  12. Cage graphs
  13. Vertex expansions of cubic graphs

Key Result

Theorem 15

Let $\mu$ be an STC of a simple connected graph $G$ with maximum degree $\Delta$, where $\beta(\mu)>0$ or $\gamma(\mu)>1$ in case $\mu$ is a TC. Let $S=\{v_0,e_1,v_1,e_2,v_2,\ldots,e_k,v_k\}$ be an MCAP of $G$ wrt $\mu$, where $\mu(S)=\{c_0,c_1\}$ and $\mu(v_0)=c_0$. Then, there exists an STC $\mu'

Figures (20)

  • Figure 1: From a lacunar STC to an equitable TC via a $\beta$-reduction for the 3-cube graph.
  • Figure 2: From a lacunar STC to an equitable TC via a $\beta$-reduction for the Heawood graph.
  • Figure 3: From a lacunar TC to an equitable TC for the Pappus graph.
  • Figure 4: From a lacunar STC to an equitable TC for the Desargues graph.
  • Figure 5: From a lacunar STC to an equitable TC for $C_8\square K_2$.
  • ...and 15 more figures

Theorems & Definitions (46)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Definition 10
  • ...and 36 more