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arXiv · 1805.05007

Square-free graphs with no six-vertex induced path

T. Karthick·Frederic Maffray
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Abstract

We elucidate the structure of $(P_6,C_4)$-free graphs by showing that every such graph either has a clique cutset, or a universal vertex, or belongs to several special classes of graphs. Using this result, we show that for any $(P_6,C_4)$-free graph $G$, $\lceil\frac{5\omega(G)}{4}\rceil$ and $\lceil\frac{\Delta(G) + \omega(G) +1}{2}\rceil$ are tight upper bounds for the chromatic number of $G$. Moreover, our structural results imply that every ($P_6$,$C_4$)-free graph with no clique cutset has bounded clique-width, and thus the existence of a polynomial-time algorithm that computes the chromatic number (or stability number) of any $(P_6,C_4)$-free graph.

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T. Karthick, Frederic Maffray. 2018-05-14. Square-free graphs with no six-vertex induced path. https://arxiv.org/abs/1805.05007

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