Euler cycles in the complete graph K2m+1 Original Research Article

We analyze the freedom one has when walking along an Euler cycle through a complete graph of an odd order: Is it possible, for any cycle C of (22m + 1) vertices, 2m + 1 of them being black, to find an edge monomorphism of C onto K2m + 1, that would be injective on the set of black vertices of C? It is shown that the answer is positive for all but two cases. Our proof is constructive, however, we relied on computers to verify approximately 37 000 cases needed for the induction basis. Our theorem generalizes a previous result on the decomposition of K2m + 1 into edge-disjoint trails of given lengths. In addition, a relation to the concept of harmonious chromatic number is mentioned.