Hajós' conjecture and cycle power graphs

Authors:
Deming Li;Mingju Liu;Yumei Peng
Affiliations:
Department of Mathematics, Capital Normal University, Beijing 100048, China;LMIB and Department of Mathematics, Beihang University, Beijing 100083, China;Department of Mathematics, Capital Normal University, Beijing 100048, China
Venue:
European Journal of Combinatorics
Year:
2010

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Abstract

Hajos' conjecture says that every graph of chromatic number k contains a subdivision of the complete graph with k vertices. In this note, we give a characterization for cycle power graphs C"n^k on Hajos' conjecture, which generalized a recent result of Thomassen (2005) [C. Thomassen, Some remarks on Hajos' conjecture, J. Combin. Theory Ser. B 93 (2005) 95105]. Precisely, we showed that for positive integers n,k such that n2k+1, and then n=q(k+1)+r, where 0@?r@?k, the kth power of the cycle C"n, C"n^k, satisfies Hajos' conjecture if and only if 1+2+...+@?r/q@?@?k.