Pushable chromatic number of graphs with degree constraints

TL;DR

This paper investigates the pushable chromatic number of oriented graphs under various degree and structural constraints, providing new bounds and exact values for specific classes of graphs, thus advancing understanding of graph coloring properties.

Contribution

It establishes new bounds on the pushable chromatic number for graphs with degree constraints and determines exact values for subcubic and certain planar graphs.

Findings

Maximum pushable chromatic number for degree 29 is between 2^{rac{\u2206}{2}-1} and ()()2^{\u2206-1}+2.

For 3, the maximum pushable chromatic number is 6 or 7.

Graphs with average degree less than 3 have pushable chromatic number between 5 and 6.

Abstract

Pushable homomorphisms and the pushable chromatic number $\chi_p$ of oriented graphs were introduced by Klostermeyer and MacGillivray in 2004. They notably observed that, for any oriented graph $\overrightarrow{G}$, we have $\chi_p(\overrightarrow{G}) \leq \chi_o(\overrightarrow{G}) \leq 2 \chi_p(\overrightarrow{G})$, where $\chi_o(\overrightarrow{G})$ denotes the oriented chromatic number of $\overrightarrow{G}$. This stands as first general bounds on $\chi_p$. This parameter was further studied in later works.This work is dedicated to the pushable chromatic number of oriented graphs fulfilling particular degree conditions. For all $\Delta \geq 29$, we first prove that the maximum value of the pushable chromatic number of an oriented graph with maximum degree $\Delta$ lies between $2^{\frac{\Delta}{2}-1}$ and $(\Delta-3) \cdot (\Delta-1) \cdot 2^{\Delta-1} + 2$ which implies an improved bound on the oriented chromatic number of the same family of graphs. For subcubic oriented graphs, that is, when $\Delta \leq 3$, we then prove that the maximum value of the pushable chromatic number is~$6$ or~$7$. We also prove that the maximum value of the pushable chromatic number of oriented graphs with maximum average degree less than~$3$ lies between~$5$ and~$6$. The former upper bound of~$7$ also holds as an upper bound on the pushable chromatic number of planar oriented graphs with girth at least~$6$.