Covering and spanning tree of graphs

In this note we recall some well-known results about covering and k-covering of graphs. Then, we introduce a new notion of spanning tree which we call the p-spanning tree. This notion allows us to reduce the upper bound for the minimum integer k for which a connected graph G has no nontrivial k-covering. This note deals with the notion of covering of graphs and more precisely with the notion of k-covering of graphs. In the rst part, we introduce the notion of covering and recall the Reidemeister theorem. In the second part, we focus on the notion of k-covering of graphs. We introduce then the p-spanning tree. 1.1 Deenition and properties The notion of graph covering is based on the notion of bijection between neighbourhoods. If we are able to nd a surjective homomorphism between two graphs G and H which maps the vertices of H onto the vertices of G with respect of adjacency, then H is a covering of G. Deenition 1 Let G be a connected graph and H be a graph, H is a covering of G if there exists a surjective homomorphism from V (H) onto V (G) such that for any v in V (H), the restriction of to N H (v) is a bijection between N H (v) and N G ((v)). Our examples use diierent geometric shapes for the vertices in order to code the homo-morphism : any vertex of the covering has the same shape as its image by the homomorphism. Example 1 The cycle with six vertices is a covering of the cycle with three vertices (cf. Figure 1). In Figure 2 we give a graph with six vertices having at least two diierent coverings with twelve vertices.