Log In Sign Up

On the dichromatic number of surfaces

by   Pierre Aboulker, et al.

In this paper, we give bounds on the dichromatic number χ⃗(Σ) of a surface Σ, which is the maximum dichromatic number of an oriented graph embeddable on Σ. We determine the asymptotic behaviour of χ⃗(Σ) by showing that there exist constants a_1 and a_2 such that, a_1√(-c)/log(-c)≤χ⃗(Σ) ≤ a_2 √(-c)/log(-c) for every surface Σ with Euler characteristic c≤ -2. We then give more explicit bounds for some surfaces with high Euler characteristic. In particular, we show that the dichromatic numbers of the projective plane ℕ_1, the Klein bottle ℕ_2, the torus 𝕊_1, and Dyck's surface ℕ_3 are all equal to 3, and that the dichromatic numbers of the 5-torus 𝕊_5 and the 10-cross surface ℕ_10 are equal to 4. We also consider the complexity of deciding whether a given digraph or oriented graph embedabble in a fixed surface is k-dicolourable. In particular, we show that for any surface, deciding whether a digraph embeddable on this surface is 2-dicolourable is NP-complete, and that deciding whether a planar oriented graph is 2-dicolourable is NP-complete unless all planar oriented graphs are 2-dicolourable (which was conjectured by Neumann-Lara).


page 1

page 2

page 3

page 4


Complexity of Domination in Triangulated Plane Graphs

We prove that for a triangulated plane graph it is NP-complete to determ...

Homomorphically Full Oriented Graphs

Homomorphically full graphs are those for which every homomorphic image ...

Complexity of planar signed graph homomorphisms to cycles

We study homomorphism problems of signed graphs. A signed graph is an un...

On the inversion number of oriented graphs

Let D be an oriented graph. The inversion of a set X of vertices in D co...

Invertibility of digraphs and tournaments

For an oriented graph D and a set X⊆ V(D), the inversion of X in D is th...

On the degree sequences of dual graphs on surfaces

Given two graphs G and G^* with a one-to-one correspondence between thei...

Computing Cliques and Cavities in Networks

Complex networks have complete subgraphs such as nodes, edges, triangles...