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A new vertex coloring heuristic and corresponding chromatic number

Manouchehr Zaker

TL;DR

This work proves that the vertices of every graph G can be effectively colored using color classes say C1,…,Ck, and proves a new coloring procedure which combines the strategies of these two techniques and satisfies an additional property.

Abstract

One method to obtain a proper vertex coloring of graphs using a reasonable number of colors is to start from any arbitrary proper coloring and then repeat some local re-coloring techniques to reduce the number of color classes. The Grundy (First-Fit) coloring and color-dominating colorings of graphs are two well-known such techniques. The color-dominating colorings are also known and commonly referred as {\rm b}-colorings. But these two topics have been studied separately in graph theory. We introduce a new coloring procedure which combines the strategies of these two techniques and satisfies an additional property. We first prove that the vertices of every graph $G$ can be effectively colored using color classes say $C_1, \ldots, C_k$ such that $(i)$ for any two colors $i$ and $j$ with $1\leq i< j \leq k$, any vertex of color $j$ is adjacent to a vertex of color $i$, $(ii)$ there exists a set $\{u_1, \ldots, u_k\}$ of vertices of $G$ such that $u_j\in C_j$ for any $j\in \{1, \ldots, k\}$ and $u_k$ is adjacent to $u_j$ for each $1\leq j \leq k$ with $j\not= k$, and $(iii)$ for each $i$ and $j$ with $i\not= j$, the vertex $u_j$ has a neighbor in $C_i$. This provides a new vertex coloring heuristic which improves both Grundy and color-dominating colorings. Denote by $z(G)$ the maximum number of colors used in any proper vertex coloring satisfying the above properties. The $z(G)$ quantifies the worst-case behavior of the heuristic. We prove the existence of $\{G_n\}_{n\geq 1}$ such that $\min \{Γ(G_n), b(G_n)\} \rightarrow \infty$ but $z(G_n)\leq 3$ for each $n$. For each positive integer $t$ we construct a family of finitely many colored graphs ${\mathcal{D}}_t$ satisfying the property that if $z(G)\geq t$ for a graph $G$ then $G$ contains an element from ${\mathcal{D}}_t$ as a colored subgraph. This provides an algorithmic method for proving numeric upper bounds for $z(G)$.

A new vertex coloring heuristic and corresponding chromatic number

TL;DR

This work proves that the vertices of every graph G can be effectively colored using color classes say C1,…,Ck, and proves a new coloring procedure which combines the strategies of these two techniques and satisfies an additional property.

Abstract

One method to obtain a proper vertex coloring of graphs using a reasonable number of colors is to start from any arbitrary proper coloring and then repeat some local re-coloring techniques to reduce the number of color classes. The Grundy (First-Fit) coloring and color-dominating colorings of graphs are two well-known such techniques. The color-dominating colorings are also known and commonly referred as {\rm b}-colorings. But these two topics have been studied separately in graph theory. We introduce a new coloring procedure which combines the strategies of these two techniques and satisfies an additional property. We first prove that the vertices of every graph can be effectively colored using color classes say such that for any two colors and with , any vertex of color is adjacent to a vertex of color , there exists a set of vertices of such that for any and is adjacent to for each with , and for each and with , the vertex has a neighbor in . This provides a new vertex coloring heuristic which improves both Grundy and color-dominating colorings. Denote by the maximum number of colors used in any proper vertex coloring satisfying the above properties. The quantifies the worst-case behavior of the heuristic. We prove the existence of such that but for each . For each positive integer we construct a family of finitely many colored graphs satisfying the property that if for a graph then contains an element from as a colored subgraph. This provides an algorithmic method for proving numeric upper bounds for .

Paper Structure

This paper contains 6 sections, 9 equations, 4 figures.

Table of Contents

  1. Introduction
  2. A new coloring heuristic
  3. Construction of $z$-atoms and their properties
  4. Results concerning trees
  5. Further researches
  6. Acknowledgment

Figures (4)

  • Figure 1: The graph $G_t$ in Proposition \ref{['infinityexample']} for $t=4$
  • Figure 2: The nine $z$-atoms from the whole eighteen triangle-free $z$-atoms with $z$-number four. For each one a $z$-coloring using four colors is illustrated, where the color-dominating vertices are indicated by circles.
  • Figure 3: The remaining nine $z$-atoms from the whole eighteen triangle-free $z$-atoms with $z$-number four. For each one a $z$-coloring using four colors is illustrated, where the color-dominating vertices are indicated by circles.
  • Figure 4: The trees $R_3$ and $R_4$ with their canonic labelings