Complexity of acyclic colorings of graphs and digraphs with degree and girth constraints

06/28/2019
by   Pavol Hell, et al.
0

We consider acyclic r-colorings in graphs and digraphs: they color the vertices in r colors, each of which induces an acyclic graph or digraph. (This includes the dichromatic number of a digraph, and the arboricity of a graph.) For any girth and sufficiently high degree, we prove the NP-completeness of acyclic r-colorings; our method also implies the known analogue for classical colorings. The proofs use high girth graphs with high arboricity and dichromatic numbers. High girth graphs and digraphs with high chromatic and dichromatic numbers have been well studied; we re-derive the results from a general result about relational systems, which also implies the similar fact about high girth and high arboricity used in the proofs. These facts concern graphs and digraphs of high girth and low degree; we contrast them by considering acyclic colorings of tournaments (which have low girth and high degree). We prove that even though acyclic two-colorability of tournaments is known to be NP-complete, random acyclically r-colorable tournaments allow recovering an acyclic r-coloring in deterministic linear time, with high probablity.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
05/03/2023

Upper Bounds on the Acyclic Chromatic Index of Degenerate Graphs

An acyclic edge coloring of a graph is a proper edge coloring without an...
research
11/05/2021

On Homomorphism Graphs

We introduce a new type of examples of bounded degree acyclic Borel grap...
research
02/03/2023

Acyclic Chromatic Index of Chordless Graphs

An acyclic edge coloring of a graph is a proper edge coloring in which t...
research
02/13/2019

On the achromatic number of signed graphs

In this paper, we generalize the concept of complete coloring and achrom...
research
11/15/2022

Acyclic colourings of graphs with obstructions

Given a graph G, a colouring of G is acyclic if it is a proper colouring...
research
02/28/2022

Improved bounds for acyclic coloring parameters

The acyclic chromatic number of a graph is the least number of colors ne...
research
12/23/2019

On the Burning Number of p-Caterpillars

The burning number is a recently introduced graph parameter indicating t...

Please sign up or login with your details

Forgot password? Click here to reset