The 3-colorability problem is a well-known NP-complete problem and it remains NP-complete for bull-free graphs, where a bull is the graph consisting of a K3 with two pendant edges attached to two of its vertices. In this paper, for k≥3, we characterize all k-colorable (bull,claw)-free graphs containing an induced cycle of length at least 6. Moreover, we present the full characterization of all non 4-colorable connected (bull,claw)-free graphs and (bull,chair,C5)-free graphs, and all non 5-colorable connected (bull,claw,C5)-free graphs.
@article{arxiv.2509.01698,
title = {On $k$-colorability of $(bull, H)$-free graphs},
author = {Nadzieja Hodur and Monika Pilśniak and Magdalena Prorok and Ingo Schiermeyer},
journal= {arXiv preprint arXiv:2509.01698},
year = {2025}
}