A Strengthening of Brooks' Theorem for Line Graphs
We prove that if $G$ is the line graph of a multigraph, then the chromatic number $\chi(G)$ of $G$ is at most $\max\left\{\omega(G), \frac{7\Delta(G) + 10}{8}\right\}$ where $\omega(G)$ and $\Delta(G)$ are the clique number and the maximum degree of $G$, respectively. Thus Brooks' Theorem holds for line graphs of multigraphs in much stronger form. Using similar methods we then prove that if $G$ is the line graph of a multigraph with $\chi(G) \geq \Delta(G) \geq 9$, then $G$ contains a clique on $\Delta(G)$ vertices. Thus the Borodin-Kostochka Conjecture holds for line graphs of multigraphs.
2005 ◽
Vol DMTCS Proceedings vol. AE,...
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2019 ◽
Vol 71
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pp. 113-129
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2016 ◽
Vol 16
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pp. 1750173
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1967 ◽
Vol 63
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pp. 679-681
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