On colorings of graphs without short cycles

Abstract The following theorem is proved: Let n, k be natural numbers, n ⩾ 3, let A be a set and A 1 , A 2 ,…, A r different decompositions of A into at most n classes. Then there exists an n-chromatic graph G = 〈 V ( G ), E ( G )〉 such that A ⊂ V ( G ), G does not contain cycles of length ⩽ k and G has just r colorings B 1 ,…, B r by n colours such that A i = B i |A, i = 1,…,r. From this theorem follows immediately existence of uniquely colorable graphs without short cycles. Further, characterizations of subgraphs of critical graphs may be given.