Previous article Next article Coloring Claw-Free Graphs with $\Delta-1$ ColorsDaniel W. Cranston and Landon RabernDaniel W. Cranston and Landon Rabernhttps://doi.org/10.1137/12088015XPDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractWe prove that every claw-free graph $G$ that does not contain a clique on $\Delta(G) \geq 9$ vertices can be $\Delta(G) - 1$ colored.1. A. Beutelspacher and P. R. Hering , Minimal graphs for which the chromatic number equals the maximal degree , Ars. Combin. , 18 ( 1984 ), pp. 201 -- 216 . Google Scholar2. O. V. Borodin, A. V. Kostochka and D. R. Woodall , List edge and list total colourings of multigraphs , J. Combin. Theory Ser. B , 71 ( 1997 ), pp. 184 -- 204 . CrossrefISIGoogle Scholar3. O. V. Borodin and A. V. Kostochka , On an upper bound of a graph's chromatic number, depending on the graph's degree and density , J. Combin. Theory Ser. B , 23 ( 1977 ), pp. 247 -- 250 . CrossrefISIGoogle Scholar4. R. L. Brooks , On colouring the nodes of a network , Math. Proc. Cambridge Philos. Soc. 37 , 1941 , pp. 194 -- 197 . CrossrefGoogle Scholar5. M. Chudnovsky and A. Ovetsky , Coloring quasi-line graphs , J. Graph Theory , 54 ( 2007 ), pp. 41 -- 50 . CrossrefISIGoogle Scholar6. M. Chudnovsky and P. Seymour , The structure of claw-free graphs , Surveys Combin. , 327 ( 2005 ), pp. 153 -- 171 . Google Scholar7. D. W. Cranston and L. Rabern, Conjectures equivalent to the Borodin-Kostochka conjecture that are a priori weaker, preprint, arXiv:1203.5380 (2012).Google Scholar8. M. Dhurandhar and Improvement Brooks' chromatic bound for a class of graphs , Discrete Math. , 42 ( 1982 ), pp. 51 -- 56 . CrossrefISIGoogle Scholar9. P. Erdös, A. L. Rubin and H. Taylor , Choosability in graphs, in Proceedings of the West Coast Conference on Combinatorics, Graph Theory and Computing , Congress. Numer. 26 , 1979 , pp. 125 -- 157 . Google Scholar10. S. Gravier and F. Maffray , Graphs whose choice number is equal to their chromatic number , J. Graph Theory , 27 ( 1998 ), pp. 87 -- 97 . CrossrefISIGoogle Scholar11. H. A. Kierstead and J. H. Schmerl , The chromatic number of graphs which induce neither $K_{1,3}$ nor $K_5-e$ , Discrete Math. , 58 ( 1986 ), pp. 253 -- 262 . CrossrefISIGoogle Scholar12. A. King, Claw-Free Graphs and Two Conjectures on $\omega$, $\Delta$, and $\chi$, Ph.D. thesis, McGill University, Montreal, Canada, 2009.Google Scholar13. A. D. King and B. A. Reed , Bounding $\chi$ in terms of $\omega$ and $\Delta$ for quasi-line graphs , J. Graph Theory , 59 ( 2008 ), pp. 215 -- 228 . CrossrefISIGoogle Scholar14. A. V. Kostochka, personal communication, 2012.Google Scholar15. A. V. Kostochka , and chromatic number , Metody Diskret. Anal. , 35 ( 1980 ), pp. 45 -- 70 (in Russian). Google Scholar16. M. Molloy and B. Reed, Graph Colouring and the Probabilistic Method, Springer, Berlin, 2002.Google Scholar17. J. A. Noel, B. A. Reed, and H. Wu, A Proof of a Conjecture of Ohba, preprint, arXiv:1211.1999 (2012).Google Scholar18. L. Rabern , A strengthening of Brooks' Theorem for line graphs , Electron. J. Combin. , 18 ( 2011 ), p. 1 . ISIGoogle Scholar19. B. Reed , A strengthening of Brooks' theorem , J. Combin. Theory Ser. B , 76 ( 1999 ), pp. 136 -- 149 . CrossrefISIGoogle ScholarKeywordscoloringlist coloringclaw-freeBorodin--Kostochka conjecture Previous article Next article FiguresRelatedReferencesCited byDetails Coloring (P5,gem) $({P}_{5},\text{gem})$‐free graphs with Δ−1 ${\rm{\Delta }}-1$ colors9 June 2022 | Journal of Graph Theory, Vol. 101, No. 4 Cross Ref Large cliques in graphs with high chromatic numberDiscrete Mathematics, Vol. 23 Cross Ref Borodin–Kostochka’s conjecture on $$(P_5,C_4)$$-free graphs4 August 2020 | Journal of Applied Mathematics and Computing, Vol. 65, No. 1-2 Cross Ref List-Coloring Claw-Free Graphs with $\Delta-1$ ColorsDaniel W. Cranston and Landon Rabern6 April 2017 | SIAM Journal on Discrete Mathematics, Vol. 31, No. 2AbstractPDF (513 KB)Brooks' Theorem and Beyond26 December 2014 | Journal of Graph Theory, Vol. 80, No. 3 Cross Ref Graphs with $\chi=\Delta$ Have Big CliquesDaniel W. Cranston and Landon Rabern1 October 2015 | SIAM Journal on Discrete Mathematics, Vol. 29, No. 4AbstractPDF (372 KB)Beyond Ohba’s Conjecture: A bound on the choice number of k -chromatic graphs with n verticesEuropean Journal of Combinatorics, Vol. 43 Cross Ref Volume 27, Issue 1| 2013SIAM Journal on Discrete Mathematics History Submitted:07 June 2012Accepted:17 December 2012Published online:14 March 2013 Information© 2013, Society for Industrial and Applied MathematicsKeywordscoloringlist coloringclaw-freeBorodin--Kostochka conjectureMSC codes05C15PDF Download Article & Publication DataArticle DOI:10.1137/12088015XArticle page range:pp. 534-549ISSN (print):0895-4801ISSN (online):1095-7146Publisher:Society for Industrial and Applied Mathematics