On polyhedral embeddings of cubic graphs Bojan Mohar Department of Mathematics University of Ljubljana 1000 Ljubljana, Slovenia bojan.mohar@uni-lj.si Andrej Vodopivec Department of Mathematics IMFM 1000 Ljubljana, Slovenia andrej.vodopivec@fmf.uni-lj.si January 29, 2004 Abstract Polyhedral embeddings of cubic graphs by means of certain oper- ations are studied. It is proved that some known families of snarks have no (orientable) polyhedral embeddings. This result supports a conjecture of Gr¨ unbaum that no snark admits an orientable polyhe- dral embedding. This conjecture is verified for all snarks having up to 30 vertices using computer. On the other hand, for every non- orientable surface S, there exists a non 3-edge-colorable graph, which polyhedrally embeds in S. Keywords: polyhedral embedding, cubic graph, snark, flower snark, Goldberg snark. 1 Introduction In this paper we study embeddings of cubic graphs in closed surfaces. We refer to [5] for basic terminology and properties of embeddings. Following the approach of [5], all embeddings are assumed to be 2-cell embeddings. Two embeddings of a graph are considered to be (combinatorially) equal, if they have the same set of facial walks. If S is a surface with Euler characteristic χ(S ), then ǫ(S ) := 2 − χ(S ) is a non-negative integer, which is called the Euler genus of S . If an embedding of a graph G in a non-orientable surface is given by a rotation system and a signature λ : E(G) →{+1, −1} and H is an acyclic subgraph of G, then we can assume that the edges of H have positive sig- nature, λ(e) = 1 for all e ∈ E(H ). We shall assume this in several instances 1