chi-bounded

Definition: For $\chi$ being chromatic number and $\omega$ being maximum clique we say a graph class is $\chi$-bounded if there exists a function $f$ such that $\chi(G) \le f(\omega(G))$ for every $G$ from the class.

Relations

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

Results

2012 Classes of graphs with small rank decompositions are χ-bounded by Dvořák, Král’
- page 2 : rank-width upper bounds chi-bounded by a constant – Theorem 1. For any $k$, the class of graphs with rank-width at most $k$ is $\chi$-bounded.
unknown source
- perfect upper bounds chi-bounded by a constant
- clique-width upper bounds chi-bounded by a constant
- series-parallel upper bounds chi-bounded by a constant
assumed
- chi-bounded is equivalent to chi-bounded – assumed