arXiv · 1905.11578
We prove that the chromatic number of a circle graph with clique number $ω$ is at most $7ω^2$.
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James Davies, Rose McCarty. 2019-05-28. Circle graphs are quadratically $χ$-bounded. https://doi.org/10.1112/blms.12447
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Under a local density condition, we prove that every $k$-edge spider embeds at any prescribed center of degree at least $k$, unless all legs are even and the host graph has one of two specified structures. These structures contain complete bipartite subgraphs with prescribed neighborhoods. The proof uses path rerouting and three exchange lemmas that describe equality in neighborhood estimates. As a consequence, we recover the Erd\H os-Sós bound for all spiders.
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