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arXiv · 2511.23195

On the structure of ($4K_1$, $C_4$, $P_6$)-free graphs

Chính T. Hoàng·Ramin Javadi·Nicolas Trotignon
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Abstract

Determining the complexity of colouring ($4K_1, C_4$)-free graph is a long open problem. Recently Penev showed that there is a polynomial-time algorithm to colour a ($4K_1, C_4, C_6$)-free graph. In this paper, we will prove that if $G$ is a ($4K_1, C_4, P_6$)-free graph that contains a $C_6$, then $G$ has bounded clique-width. To this purpose, we use a new method to bound the clique-width, that is of independent interest. As a consequence, there is a polynomial-time algorithm to colour ($4K_1, C_4, P_6$)-free graphs.

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Chính T. Hoàng, Ramin Javadi, Nicolas Trotignon. 2025-11-28. On the structure of ($4K_1$, $C_4$, $P_6$)-free graphs. https://arxiv.org/abs/2511.23195

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