# A new vertex coloring heuristic and corresponding chromatic number

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 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, …, C_k such that (i) for any two colors i and j with 1≤ i< j ≤ k, any vertex of color j is adjacent to a vertex of color i, (ii) there exists a set {u_1, …, u_k} of vertices of G such that u_j∈ C_j for any j∈{1, …, k} and u_k is adjacent to u_j for each 1≤ j ≤ k with j≠ k, and (iii) for each i and j with i≠ 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≥ 1 such that min{Γ(G_n), b(G_n)}→∞ but z(G_n)≤ 3 for each n. For each positive integer t we construct a family of finitely many colored graphs 𝒟_t satisfying the property that if z(G)≥ t for a graph G then G contains an element from 𝒟_t as a colored subgraph. This provides an algorithmic method for proving numeric upper bounds for z(G).

research
02/02/2023

### More results on the z-chromatic number of graphs

By a z-coloring of a graph G we mean any proper vertex coloring consisti...
research
08/02/2019

### b-continuity and Partial Grundy Coloring of graphs with large girth

A b-coloring of a graph is a proper coloring such that each color class ...
research
07/22/2023

### Total Domination, Separated Clusters, CD-Coloring: Algorithms and Hardness

Domination and coloring are two classic problems in graph theory. The ma...
research
02/11/2020

### Local WL Invariance and Hidden Shades of Regularity

The k-dimensional Weisfeiler-Leman algorithm (k-WL) is a powerful tool f...
research
09/19/2022

### Gradual Weisfeiler-Leman: Slow and Steady Wins the Race

The classical Weisfeiler-Leman algorithm aka color refinement is fundame...
research
02/19/2021

### A matheuristic approach for the b-coloring problem using integer programming and a multi-start multi-greedy randomized metaheuristic

Given a graph G=(V,E), the b-coloring problem consists in attributing a ...
research
01/29/2022

### New results on the robust coloring problem

Many variations of the classical graph coloring model have been intensiv...

Please sign up or login with your details

Forgot password? Click here to reset