acyclic chromatic number

providers: ISGCI

Definition: Minimum number of colors such that there is a proper coloring and the graph induced on any two colors is acyclic.

Relations

Other		Relation from	Relation to
acyclic chromatic number	yellow■	equal	equal
admissibility	gray■	unknown to HOPS	unknown to HOPS
arboricity	red■	exclusion	upper bound
average degree	red■	exclusion	upper bound
average distance	blue■	exclusion	exclusion
bandwidth	green■	upper bound	exclusion
bipartite	blue■	unbounded	exclusion
bipartite number	blue■	exclusion	exclusion
bisection bandwidth	blue■	exclusion	exclusion
block	blue■	unbounded	exclusion
book thickness	lime■	upper bound	unknown to HOPS
boolean width	blue■	exclusion	exclusion
bounded components	green■	upper bound	exclusion
bounded expansion	gray■	unknown to HOPS	unknown to HOPS
boxicity	red■	exclusion	upper bound
branch width	green■	upper bound	exclusion
c-closure	blue■	exclusion	exclusion
carving-width	green■	upper bound	exclusion
chi-bounded	magenta■	exclusion	unknown to HOPS
chordal	blue■	unbounded	exclusion
chordality	red■	exclusion	upper bound
chromatic number	red■	exclusion	upper bound
clique cover number	blue■	exclusion	exclusion
clique-tree-width	blue■	exclusion	exclusion
clique-width	blue■	exclusion	exclusion
cluster	blue■	unbounded	exclusion
co-cluster	blue■	unbounded	exclusion
cograph	blue■	unbounded	exclusion
complete	blue■	unbounded	exclusion
connected	blue■	exclusion	avoids
contraction complexity	green■	upper bound	exclusion
cutwidth	green■	upper bound	exclusion
cycle	green■	upper bound	exclusion
cycles	green■	upper bound	exclusion
d-admissibility	gray■	unknown to HOPS	unknown to HOPS
d-path-free	green■	upper bound	exclusion
degeneracy	red■	exclusion	upper bound
degree treewidth	green■	upper bound	exclusion
diameter	blue■	exclusion	exclusion
diameter+max degree	green■	upper bound	exclusion
distance to bipartite	blue■	exclusion	exclusion
distance to block	blue■	exclusion	exclusion
distance to bounded components	green■	upper bound	exclusion
distance to chordal	blue■	exclusion	exclusion
distance to cluster	blue■	exclusion	exclusion
distance to co-cluster	blue■	exclusion	exclusion
distance to cograph	blue■	exclusion	exclusion
distance to complete	blue■	exclusion	exclusion
distance to edgeless	green■	upper bound	exclusion
distance to forest	green■	upper bound	exclusion
distance to interval	blue■	exclusion	exclusion
distance to linear forest	green■	upper bound	exclusion
distance to maximum degree	green■	upper bound	exclusion
distance to outerplanar	green■	upper bound	exclusion
distance to perfect	blue■	exclusion	exclusion
distance to planar	green■	upper bound	exclusion
distance to stars	green■	upper bound	exclusion
domatic number	red■	exclusion	upper bound
domination number	blue■	exclusion	exclusion
domino treewidth	green■	upper bound	exclusion
edge clique cover number	blue■	exclusion	exclusion
edge connectivity	red■	exclusion	upper bound
edge-cut width	green■	upper bound	exclusion
edge-treewidth	green■	upper bound	exclusion
edgeless	green■	upper bound	avoids
excluded minor	gray■	unknown to HOPS	unknown to HOPS
excluded planar minor	green■	upper bound	avoids
excluded top-minor	gray■	unknown to HOPS	unknown to HOPS
feedback edge set	green■	upper bound	exclusion
feedback vertex set	green■	upper bound	exclusion
flip-width	magenta■	exclusion	unknown to HOPS
forest	green■	upper bound	exclusion
genus	green■	upper bound	exclusion
grid	green■	upper bound	exclusion
h-index	green■	upper bound	exclusion
interval	blue■	unbounded	exclusion
iterated type partitions	blue■	exclusion	exclusion
linear clique-width	blue■	exclusion	exclusion
linear forest	green■	upper bound	exclusion
linear NLC-width	blue■	exclusion	exclusion
linear rank-width	blue■	exclusion	exclusion
maximum clique	red■	exclusion	upper bound
maximum degree	green■	upper bound	exclusion
maximum independent set	blue■	exclusion	exclusion
maximum induced matching	blue■	exclusion	exclusion
maximum leaf number	green■	upper bound	exclusion
maximum matching	green■	upper bound	exclusion
maximum matching on bipartite graphs	green■	upper bound	exclusion
merge-width	magenta■	exclusion	unknown to HOPS
mim-width	magenta■	exclusion	unknown to HOPS
minimum degree	red■	exclusion	upper bound
mm-width	green■	upper bound	exclusion
modular-width	blue■	exclusion	exclusion
module-width	blue■	exclusion	exclusion
monadically dependent	magenta■	exclusion	unknown to HOPS
monadically stable	gray■	unknown to HOPS	unknown to HOPS
neighborhood diversity	blue■	exclusion	exclusion
NLC-width	blue■	exclusion	exclusion
NLCT-width	blue■	exclusion	exclusion
nowhere dense	gray■	unknown to HOPS	unknown to HOPS
odd cycle transversal	blue■	exclusion	exclusion
outerplanar	green■	upper bound	exclusion
overlap treewidth	green■	upper bound	exclusion
path	green■	upper bound	exclusion
pathwidth	green■	upper bound	exclusion
pathwidth+maxdegree	green■	upper bound	exclusion
perfect	blue■	unbounded	exclusion
planar	green■	upper bound	exclusion
radius-inf flip-width	blue■	exclusion	exclusion
radius-r flip-width	magenta■	exclusion	unknown to HOPS
rank-width	blue■	exclusion	exclusion
series-parallel	gray■	unknown to HOPS	unknown to HOPS
shrub-depth	blue■	exclusion	exclusion
sim-width	magenta■	exclusion	unknown to HOPS
size	green■	upper bound	exclusion
slim tree-cut width	green■	upper bound	exclusion
sparse twin-width	cyan■	unknown to HOPS	exclusion
star	green■	upper bound	exclusion
stars	green■	upper bound	exclusion
strong coloring number	gray■	unknown to HOPS	unknown to HOPS
strong d-coloring number	gray■	unknown to HOPS	unknown to HOPS
strong inf-coloring number	green■	upper bound	exclusion
topological bandwidth	green■	upper bound	exclusion
tree	green■	upper bound	exclusion
tree-cut width	green■	upper bound	exclusion
tree-independence number	magenta■	exclusion	unknown to HOPS
tree-partition-width	green■	upper bound	exclusion
treebandwidth	green■	upper bound	exclusion
treedepth	green■	upper bound	exclusion
treelength	magenta■	exclusion	unknown to HOPS
treespan	green■	upper bound	exclusion
treewidth	green■	upper bound	exclusion
twin-cover number	blue■	exclusion	exclusion
twin-width	blue■	exclusion	exclusion
vertex connectivity	gray■	unknown to HOPS	unknown to HOPS
vertex cover	green■	upper bound	exclusion
vertex integrity	green■	upper bound	exclusion
weak coloring number	gray■	unknown to HOPS	unknown to HOPS
weak d-coloring number	gray■	unknown to HOPS	unknown to HOPS
weak inf-coloring number	green■	upper bound	exclusion
weakly sparse	red■	exclusion	upper bound
weakly sparse and merge width	gray■	unknown to HOPS	unknown to HOPS

Results

2019 The Graph Parameter Hierarchy by Sorge
- page 3 : acyclic chromatic number – The \emph{acyclic chromatic number} of a graph $G = (V,E)$ is the smallest size of a vertex partition $P={V_1,\dots,V_\ell}$ such that each $V_i$ is an independent set and for all $V_i,V_j$ the graph $G[V_i \cup V_j]$ does not contain a cycle.
- page 8 : maximum degree upper bounds acyclic chromatic number by a polynomial function – Lemma 4.6 ([15]). The acyclic chromatic number $\chi_a$ is uppre bounded by the maximum degree $\Delta$ (for every graph with $\Delta > 4$). We have $\chi_a \le \Delta(\Delta-1)/2$.
- page 8 : h-index upper bounds acyclic chromatic number by a polynomial function – Lemma 4.7. The acyclic chromatic number $\chi_a$ is upper bounded by the $h$-index $h$. We have $\chi_a \le h(h+1)/2$.
- page 8 : genus upper bounds acyclic chromatic number by a linear function – Lemma 4.8 ([3]). The accylic chromatic number $\chi_a$ is upper bounded by the genus $g$. We have $\chi_a \le 4g+4$.
- page 8 : acyclic chromatic number upper bounds boxicity by a computable function – Lemma 4.9. The boxicity $b$ is upper bounded by the acyclic chromatic number $\chi_a$ (for every graph with $\chi_a>1$). We have $b \le \chi_a(\chi_a-1)$.
- page 9 : acyclic chromatic number upper bounds degeneracy by a polynomial function – Lemma 4.18. The acyclic chromatic number $a$ upper bounds the degeneracy $d$. We have $d \le 2 \binom a2 - 1$
2004 Track Layouts of Graphs by Dujmović, Pór, Wood
- page 14 : book thickness upper bounds acyclic chromatic number by a computable function – Theorem 5. Acyclic chromatic number is bounded by stack-number (ed: a.k.a. book-thickness). In particular, every $k$-stack graph $G$ has acyclich chromatic number $\chi_a(G) \le 80^{k(2k-1)}$.
unknown source
- acyclic chromatic number upper bounds boxicity by a computable function
- book thickness upper bounds acyclic chromatic number by a computable function
- distance to planar upper bounds acyclic chromatic number by a computable function
assumed
- acyclic chromatic number is equivalent to acyclic chromatic number – assumed