Acyclic dichromatic number of oriented graphs
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911286095249408 |
|---|---|
| author | Bang-Jensen, Jørgen Picasarri-Arrieta, Lucas Yeo, Anders |
| author_facet | Bang-Jensen, Jørgen Picasarri-Arrieta, Lucas Yeo, Anders |
| contents | The dichromatic number $\vecχ(D)$ of a digraph $D=(V,A)$ is the minimum number of sets in a partition $V_1,\ldots{},V_k$ of $V$ into $k$ subsets so that the induced subdigraph $D[V_i]$ is acyclic for each $i\in [k]$. This is a generalization of the chromatic number for undirected graphs as a graph has chromatic number at most $k$ if and only if the complete biorientation of $G$ (replace each edge by a directed 2-cycle) has dichromatic number at most $k$. In this paper we introduce the acyclic dichromatic number $\vecχ_{\rm a}(D)$ of a digraph $D$ as the minimum number of sets in a partition $V_1,\ldots{},V_k$ of $V$ so that the induced subdigraph $D[V_i]$ is acyclic for each $i\in [k]$ and each of the bipartite induced subdigraphs $D[V_i,V_j]$ is acyclic for each $1\leq i<j\leq k$. This parameter, which resembles the definition of acyclic chromatic number for undirected graphs, has apparently not been studied before.
We derive a number of results which display the difference between the dichromatic number and the acyclic dichromatic number, in particular, there are digraphs $D$ with arbitrarily large $\vecχ_{\rm a}(D)-\vecχ(D)$, even among tournaments with dichromatic number 2 and bipartite tournaments (where the dichromatic number is always 2). We prove several complexity results, including that deciding whether $\vecχ_{\rm a}(D)\leq 2$ is NP-complete already for bipartite digraphs, while it is polynomial for tournaments (contrary to the case for dichromatic number). We also generalize the concept of heroes of a tournament to acyclic heroes of tournaments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_20246 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Acyclic dichromatic number of oriented graphs Bang-Jensen, Jørgen Picasarri-Arrieta, Lucas Yeo, Anders Combinatorics Discrete Mathematics The dichromatic number $\vecχ(D)$ of a digraph $D=(V,A)$ is the minimum number of sets in a partition $V_1,\ldots{},V_k$ of $V$ into $k$ subsets so that the induced subdigraph $D[V_i]$ is acyclic for each $i\in [k]$. This is a generalization of the chromatic number for undirected graphs as a graph has chromatic number at most $k$ if and only if the complete biorientation of $G$ (replace each edge by a directed 2-cycle) has dichromatic number at most $k$. In this paper we introduce the acyclic dichromatic number $\vecχ_{\rm a}(D)$ of a digraph $D$ as the minimum number of sets in a partition $V_1,\ldots{},V_k$ of $V$ so that the induced subdigraph $D[V_i]$ is acyclic for each $i\in [k]$ and each of the bipartite induced subdigraphs $D[V_i,V_j]$ is acyclic for each $1\leq i<j\leq k$. This parameter, which resembles the definition of acyclic chromatic number for undirected graphs, has apparently not been studied before. We derive a number of results which display the difference between the dichromatic number and the acyclic dichromatic number, in particular, there are digraphs $D$ with arbitrarily large $\vecχ_{\rm a}(D)-\vecχ(D)$, even among tournaments with dichromatic number 2 and bipartite tournaments (where the dichromatic number is always 2). We prove several complexity results, including that deciding whether $\vecχ_{\rm a}(D)\leq 2$ is NP-complete already for bipartite digraphs, while it is polynomial for tournaments (contrary to the case for dichromatic number). We also generalize the concept of heroes of a tournament to acyclic heroes of tournaments. |
| title | Acyclic dichromatic number of oriented graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2511.20246 |