Finiteness of k-vertex-critical graphs holds in (P4+ℓP1, chair)-free, (P4+ℓP1,P5,bull)-free, (P4+ℓP1,P5,cricket)-free, and more generally (P4+ℓP1,B4(m),B3(m)+)-free graphs, with χ ≤ ℓ+2 for (P4+ℓP1,K3)-free graphs.
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Full complexity classification for three b- and fall-coloring problems in H-free graphs plus a separation showing b-Chromatic Number can be NP-hard while Tight b-Chromatic Number is P-time solvable for some H.
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Vertex-critical graphs in subfamilies of $(P_4+\ell P_1)$-free graphs
Finiteness of k-vertex-critical graphs holds in (P4+ℓP1, chair)-free, (P4+ℓP1,P5,bull)-free, (P4+ℓP1,P5,cricket)-free, and more generally (P4+ℓP1,B4(m),B3(m)+)-free graphs, with χ ≤ ℓ+2 for (P4+ℓP1,K3)-free graphs.
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Optimal b-Colourings and Fall Colourings in $H$-Free Graphs
Full complexity classification for three b- and fall-coloring problems in H-free graphs plus a separation showing b-Chromatic Number can be NP-hard while Tight b-Chromatic Number is P-time solvable for some H.