Uniquely <i>D</i>-colourable Digraphs with Large Girth

Harutyunyan, Ararat; Kayll, P. Mark; Mohar, Bojan; Rafferty, Liam

doi:10.4153/cjm-2011-084-9

Cited by 7 publications

(24 citation statements)

References 30 publications

(73 reference statements)

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“…The interest in cores comes from their role as minimal representatives of homomorphic equivalence classes of graphs. In [HKMR11], the following corresponding definition of digraph cores was considered: A digraph D is called a core, if every acyclic homomorphism φ : V (D) → V (D) of D to itself is a bijection. Similarly, we now define a digraph D to be a circular core if any circular homomorphism φ : V (D) → V (D) is bijective.…”

Section: Conclusive Remarksmentioning

confidence: 99%

On the Complexity of Digraph Colourings and Vertex Arboricity

Hochstättler,

Schröder,

Steiner

2018

Preprint

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It has been shown by Bokal et al. that deciding 2-colourability of digraphs is an NPcomplete problem. This result was later on extended by Feder et al. to prove that deciding whether a digraph has a circular p-colouring is NP-complete for all rational p > 1. In this paper, we consider the complexity of corresponding decision problems for related notions of fractional colourings for digraphs and graphs, including the star dichromatic number, the fractional dichromatic number and the circular vertex arboricity. We prove that deciding wether a digraph admits star dichromatic number at most p resp. whether a graph has circular vertex arboricity at most p is NP-complete for every rational number p > 1 while deciding whether a digraph has fractional dichromatic number at most p is NP-complete for every real number p > 1, p = 2. To show these results, different techniques are required in each case. For the purpose of proving the first result, we relate the star dichromatic number to a new notion of homomorphisms between digraphs, called circular homomorphisms, which might be of independent interest. We provide a classification of the computational complexities of the corresponding homomorphism colouring problems similar to the one derived in [FHM03] for acyclic homomorphisms.

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Section: Conclusive Remarksmentioning

confidence: 99%

On the Complexity of Digraph Colourings and Vertex Arboricity

Hochstättler,

Schröder,

Steiner

2018

Preprint

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“…In the present article, we follow the flow, from colourings to homomorphisms and from graphs to digraphs. This work is a sequel to [8], with which we assume some familiarity. For example, because the Introduction of [8] is more extensive than this one, we refer the reader there for more background.…”

Section: Introductionmentioning

confidence: 99%

“…This work is a sequel to [8], with which we assume some familiarity. For example, because the Introduction of [8] is more extensive than this one, we refer the reader there for more background. Also, some of the arguments from [8]-e.g.…”

Section: Introductionmentioning

confidence: 99%

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Uniquely $D$-Colourable Digraphs with Large Girth II: Simplification via Generalization

Kayll¹,

Parsa²

2021

Electron. J. Combin.

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We prove that for every digraph $D$ and every choice of positive integers $k$, $\ell$ there exists a digraph $D^*$ with girth at least $\ell$ together with a surjective acyclic homomorphism $\psi\colon D^*\to D$ such that: (i) for every digraph $C$ of order at most $k$, there exists an acyclic homomorphism $D^*\to C$ if and only if there exists an acyclic homomorphism $D\to C$; and (ii) for every $D$-pointed digraph $C$ of order at most $k$ and every acyclic homomorphism $\varphi\colon D^*\to C$ there exists a unique acyclic homomorphism $f\colon D\to C$ such that $\varphi=f\circ\psi$. This implies the main results in [A. Harutyunyan et al., Uniquely $D$-colourable digraphs with large girth, Canad. J. Math., 64(6) (2012), 1310—1328; MR2994666] analogously with how the work [J. Nešetřil and X. Zhu, On sparse graphs with given colorings and homomorphisms, J. Combin. Theory Ser. B, 90(1) (2004), 161—172; MR2041324] generalizes and extends [X. Zhu, Uniquely $H$-colorable graphs with large girth, J. Graph Theory, 23(1) (1996), 33—41; MR1402136].

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“…Directed acyclic chromatic number was investigated and the analogy of Theorem 1 for directed acyclic coloring immediately follows from the random placement method (for a different proof see [9]). The existence of uniquely acyclic colorable was proved recently in [34].…”

Section: No (Transfinite) Unbounded χmentioning

confidence: 99%

A Combinatorial Classic — Sparse Graphs with High Chromatic Number

Nešetřil

2013

Bolyai Society Mathematical Studies

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Uniquely D-colourable Digraphs with Large Girth

Cited by 7 publications

References 30 publications

On the Complexity of Digraph Colourings and Vertex Arboricity

On the Complexity of Digraph Colourings and Vertex Arboricity

Uniquely $D$-Colourable Digraphs with Large Girth II: Simplification via Generalization

A Combinatorial Classic — Sparse Graphs with High Chromatic Number

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