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

## TL;DR

This paper proves new properties of b-colorings and partial Grundy colorings in graphs with large girth, extending known results and establishing b-continuity for graphs with girth at least 8.

## Contribution

It establishes that graphs with girth at least 8 are b-continuous and characterizes the b-spectrum for graphs with girth at least 7, also relating the partial Grundy number to known bounds.

## Key findings

- Graphs with girth ≥ 8 are b-continuous.
- The b-spectrum of graphs with girth ≥ 7 includes integers from 2χ(G) to b(G).
- The partial Grundy number equals a known upper bound for graphs with girth ≥ 7.

## Abstract

A b-coloring of a graph is a proper coloring such that each color class has at least one vertex which is adjacent to each other color class. The b-spectrum of $G$ is the set $S_{b}(G)$ of integers $k$ such that $G$ has a b-coloring with $k$ colors and $b(G)=\max S_{b}(G)$ is the b-chromatic number of $G$. A graph is b-continous if $S_{b}(G)=[\chi(G),b(G)]\cap \mathbb{Z}$. An infinite number of graphs that are not b-continuous is known. It is also known that graphs with girth at least 10 are b-continuous. A partial Grundy coloring is a proper coloring $f:V(G)\rightarrow \{1,\ldots,k\}$ such that each color class $i$ contains some vertex $u$ that is adjacent to every color class $j$ such that $j<i$. The partial Grundy number of $G$ is the maximum value $\partial\Gamma(G)$ for which $G$ has a partial Grundy coloring. In this work, we prove that graphs with girth at least 8 are b-continuous, and that the b-spectrum of a graph $G$ with girth at least 7 contains the integers between $2\chi(G)$ and $b(G)$. We also prove that $\partial\Gamma(G)$ equals a known upper bound when $G$ is a graph with girth at least 7. These results generalize previous ones by Linhares-Sales and Silva (2017), and by Shi et al.(2005).