# Critical H-free graphs

This page contains complete lists of critical H-free graphs. An H-free graph is a graph that does not contain H as an induced subgraph. A graph G is k-chromatic if it is k-colourable but not (k-1)-colourable. We say that a graph G is k-critical H-free if G is H-free, k-chromatic, and every H-free proper subgraph of G is (k-1)-colourable.

In  and  it is described how the following complete lists of critical H-free graphs were obtained. The program CriticalPfreeGraphs which was used to generate these lists can be downloaded here.

The graph lists are currently only available in 'graph6' format. We abbreviate the set of (k+1)-critical (Pt,Cu)-free graphs as NkPtCu and the set of (k+1)-critical (Pt+Pu)-free graphs as NkPt+Pu (where Pt is a path with t vertices and Pt+Pu stands for the disjoint union of a Pt and a Pu).

Case Critical graphs Vertex-critical graphs
N3P6 24 80
planar N3P7 52 462
N3P7C4 17 35
N3P7C5 6 27
N3P8C4 94 164
N3P3+P2 17 50
N3P4+P1 4 9

In  it is proven that there are infinitely many 4-critical P7-free graphs. Nevertheless, in Table 2 of  all 4-critical and 4-vertex-critical P7-free graphs were determined for small orders. More specifically, there are exactly 2608 4-critical and exactly 62126 4-vertex-critical P7-free graphs with at most 16 vertices.

## Obstructions for list 3-colouring

Let G be a graph and let L be a mapping that maps each vertex of G to a subset of {1,...,k}. We say that the pair (G,L) is colourable if there is a proper colouring c of G with c(v) ∈ L(v) for each v ∈ V(G). To this end, a pair (G,L) with L(v) ⊆ {1,...,k} is called a list-obstruction for k-colourability if (G,L) is not colourable. If, moreover, for all proper induced subgraphs H of G the pair (H,LV(H)) is colourable, we call (G,L) a minimal list-obstruction. (Here LV(H) denotes the restriction of L to the domain V(H)).

The table below contains the counts of all P6-free minimal list-obstructions for 3-colourability up to 9 vertices. The obstructions up to 8 vertices can also be downloaded as adjacency lists. In  it was proven that there are only finitely many P6-free minimal list-obstructions for 3-colourability, but the exact number of obstructions is not known. The program ListCriticalPfreeGraphs which was used to generate these minimal list-obstructions can be downloaded here.

Vertices No. of list-obstructions
11
21
34
443
5117
61806
734721
8196231
91208483
≥ 10?

## References

 M. Chudnovsky, J. Goedgebeur, O. Schaudt, and M. Zhong, Obstructions for three-coloring graphs with one forbidden induced subgraph, Proc. Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA16), Arlington, Virginia, USA, pages 1774-1783, 2016. Preprint: arXiv:1504.06979
 J. Goedgebeur and O. Schaudt, Journal of Graph Theory, 87(2):188-207, 2018.
 M. Chudnovsky, J. Goedgebeur, O. Schaudt, and M. Zhong, Obstructions for three-coloring and list three-coloring H-free graphs, to appear in SIAM Journal on Discrete Mathematics, 40 pages, 2019.