Hajós’ conjecture and cycle power graphs

Hajós’ 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 Hajós’ conjecture, which generalized a recent result of Thomassen (2005) [C. Thomassen, Some remarks on Hajós’ conjecture, J. Combin. Theory Ser. B 93 (2005) 95105]. Precisely, we showed that for positive integers n , k such that n > 2 k + 1 , and then n = q ( k + 1 ) + r , where 0 ≤ r ≤ k , the k th power of the cycle C n , C n k , satisfies Hajós’ conjecture if and only if 1 + 2 + ⋯ + ⌈ r / q ⌉ ≤ k .