Turán's Theorem and Maximal Degrees


Extending results of Bollobás and Thomason (1981,J. Combin. Theory Ser. B31, 111-114) and Bondy (1983,J. Combin. Theory Ser. B34, 109-111), we characterize graphs of ordernand size at leasttr(n) that do not have a vertexxof maximal degreedxwhose neighbours span at leasttr-1(dx)+1 edges. Furthermore, we show that, for every graphGof ordernand size at leasttr(n), the degree-greedy algorithm used by Bondy (1983,J. Combin. Theory Ser. B34, 109-111) and Bollobás and Thomason (1985,Ann. Discr. Math.28, 47-97) constructs a complete graphKr+1, unlessGis the Turán graphTr(n). © 1999 Academic Press.

Publication Title

Journal of Combinatorial Theory. Series B