05569118

j

05569118 Borodin, O.V. Glebov, A.N. Montassier, M. Raspaud, A. Planar graphs without 5- and 7-cycles and without adjacent triangles are 3-colorable. J. Comb. Theory, Ser. B 99, No. 4, 668-673 (2009). 2009 Elsevier Science (Academic Press), San Diego, CA EN planar graphs coloring three color problem Steinberg's conjecture Zbl 1056.05052 Zbl 1108.05046

doi:10.1016/j.jctb.2008.11.001

{\it 0. V. Borodin}, {\it A. N. Glebvov}, {\it A. Raspaud} and {\it M. R. Salavatipour} [J. Comb. Theory, Ser. B 93, No. 2, 303--311 (2005; Zbl 1056.05052)] showed that planar graphs without cycles of length 4, 5, 6, or 7 are 3-colorable. The present authors improve this result by proving that planar graphs without cycles of length 5 or 7 and also without adjacent triangles are 3-colorable. They also give counterexamples to the argument given for the same result by {\it B. Xu} [J. Comb. Theory, Ser. B 96, No. 6, 958--963 (2006; Zbl 1108.05046)]. Arthur T. White (Kalamazoo)