Parameter: acyclic chromatic number

Definition:

The acyclic chromatic number of a graph $G$ is the smallest size of a vertex partition $\{V_1,\dots,V_l\}$ such that each $V_i$ is an independent set and for all $i,j$ that graph $G[V_i\cup V_j]$ does not contain a cycle.

References

[1740]
B. Grünbaum
Acyclic colorings of planar graphs
Israel J. of Math. 14 No.4 390-408 (1973)

Relations

Minimal/maximal is with respect to the contents of ISGCI. Only references for direct bounds are given. Where no reference is given, check equivalent parameters.

Minimal upper bounds for this parameter

[+]Details

This parameter is a minimal upper bound for

[+]Details

Problems

Problems in italics have no summary page and are only listed when ISGCI contains a result for the current parameter.

3-Colourability
[?]
Input: A graph G in this class.
Output: True iff each vertex of G can be assigned one colour out of 3 such that whenever two vertices are adjacent, they have different colours.
Unknown to ISGCI [+]Details
Clique
[?]
Input: A graph G in this class and an integer k.
Output: True iff G contains a set S of pairwise adjacent vertices, with |S| >= k.
XP [+]Details
Clique cover
[?]
Input: A graph G in this class and an integer k.
Output: True iff the vertices of G can be partitioned into k sets Si, such that whenever two vertices are in the same set Si, they are adjacent.
Unknown to ISGCI [+]Details
Colourability
[?]
Input: A graph G in this class and an integer k.
Output: True iff each vertex of G can be assigned one colour out of k such that whenever two vertices are adjacent, they have different colours.
Unknown to ISGCI [+]Details
Domination
[?]
Input: A graph G in this class and an integer k.
Output: True iff G contains a set S of vertices, with |S| <= k, such that every vertex in G is either in S or adjacent to a vertex in S.
Unknown to ISGCI [+]Details
Feedback vertex set
[?]
Input: A graph G in this class and an integer k.
Output: True iff G contains a set S of vertices, with |S| <= k, such that every cycle in G contains a vertex from S.
Unknown to ISGCI [+]Details
Graph isomorphism
[?]
Input: Graphs G and H in this class
Output: True iff G and H are isomorphic.
Unknown to ISGCI [+]Details
Hamiltonian cycle
[?]
Input: A graph G in this class.
Output: True iff G has a simple cycle that goes through every vertex of the graph.
Unknown to ISGCI [+]Details
Hamiltonian path
[?]
Input: A graph G in this class.
Output: True iff G has a simple path that goes through every vertex of the graph.
Unknown to ISGCI [+]Details
Independent set
[?]
Input: A graph G in this class and an integer k.
Output: True iff G contains a set S of pairwise non-adjacent vertices, such that |S| >= k.
Unknown to ISGCI [+]Details
Maximum cut
[?]
(decision variant)
Input: A graph G in this class and an integer k.
Output: True iff the vertices of G can be partitioned into two sets A,B such that there are at least k edges in G with one endpoint in A and the other endpoint in B.
Unknown to ISGCI [+]Details
Monopolarity
[?]
Input: A graph G in this class.
Output: True iff G is monopolar.
Unknown to ISGCI [+]Details
Polarity
[?]
Input: A graph G in this class.
Output: True iff G is polar.
Unknown to ISGCI [+]Details
Weighted clique
[?]
Input: A graph G in this class with weight function on the vertices and a real k.
Output: True iff G contains a set S of pairwise adjacent vertices, such that the sum of the weights of the vertices in S is at least k.
XP [+]Details
Weighted feedback vertex set
[?]
Input: A graph G in this class with weight function on the vertices and a real k.
Output: True iff G contains a set S of vertices, such that the sum of the weights of the vertices in S is at most k and every cycle in G contains a vertex from S.
Unknown to ISGCI [+]Details
Weighted independent set
[?]
Input: A graph G in this class with weight function on the vertices and a real k.
Output: True iff G contains a set S of pairwise non-adjacent vertices, such that the sum of the weights of the vertices in S is at least k.
Unknown to ISGCI [+]Details

Graph classes

Bounded

Unbounded

Open

Unknown to ISGCI