Tomaz Pisanski, University of Ljubljana, Slovenia. Thomas W. Tucker, Colgate University. Arjana Zitnik, University of Ljubljana, Slovenia

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1 Eulerian Embeddings of Graphs Tomaz Pisanski, University of Ljubljana, Slovenia Thomas W. Tucker, Colgate University Arjana Zitnik, University of Ljubljana, Slovenia Abstract A straight-ahead walk in an embedded Eulerian graph G always passes from an edge to the opposite edge adjacent to the same vertex. A straight-ahead walk is called Eulerian if all the edges of the embedded graph G are traversed in this way starting from an arbitrary edge. The embedding that contains an Eulerian straight-ahead walk is called an Eulerian embedding. In this article, we characterize some properties of Eulerian embeddings of graphs and of embeddings of graphs such that the corresponding medial graph is Eulerian embedded. There are some interesting consequences. For example, we show that every 2-edge connected graph has an orientable embedding in which no face is adjacent to itself at and edge. We also show that every connected 4-valent graph has an embedding in an orientable surface in which it is a medial graph of some graph. 1 Introduction A projection of a link on the plane denes a planar four-valent graph. Using the properties of the embedding of this graph it is possible to determine the number of components of the link and in particular, if the link in question is a knot. Our aim is to identify the property of the embedded graph which makes the separation of knots from links possible. A straight-ahead walk or a SAW in the embedded Eulerian graph G always passes from an edge to the opposite edge adjacent to the same vertex. A projection of a link in the plane is a knot if and only if it contains exactly one SAW. In this paper we assume the graphs to be nite and connected and the embeddings to be 2-cell. Let us now introduce some denitions. Graph Theory, Combinatorics, Algorithms, and Applications,

2 2 T. Pisanski, T. Tucker and A. Zitnik A circuit is a closed walk with no repeated edges. The straight ahead walks, the SAWs, of an embedded Eulerian graph G induce a circuit partition of the edges. Let us denote by s(g; S) the number of components of SAW decomposition of G. An embedding of Eulerian graph G in surface S is Eulerian, if it contains exactly one SAW, ie. s(g; S) = 1: An Eulerian directed graph is a graph together with a directed Eulerian circuit in the graph. At an Eulerian directed graph an embedding is alternating if the directions of edges alternate in the rotations at each vertex. The medial graph of an embedded graph G, M e(g), is a graph, embedded in the same surface as G and it is obtained from G as follows: the vertices of Me(G) are the edges of G and two vertices of Me(G) are adjacent if they share a common angle in G: Note that embedded graphs, which are dual to each other, have the same medial graphs. The medial graph of any graph is 4-valent and thus Eulerian. A graph is Eulerian medial embedded if its medial is Eulerian embedded. For motivation, let us pose the following question: Question 1 Does every 2-edge connected graph have an orientable embedding in which no face is adjacent to itself at a vertex? If this question were solved, it would answer the Question 2, which is its weaker version, and also the famous double cover conjecture. In the sequel we will give a positive answer to the Question 2 for Eulerian graphs. Question 2 Does every 2-edge connected graph have an orientable embedding in which no face is adjacent to itself at an edge? Question 3 The double cover conjecture: can every graph be covered with cycles such that each edge lies on exactly two cycles? Here we state a Theorem, which is obvious, but has some interesting consequences. Theorem 1 In an alternating orientable embedding of an Eulerian directed graph all face boundaries are directed consistently by edge directions. Corollary 2 An alternating orientable embedding of an Eulerian directed graph has bipartite dual.

3 Eulerian Embeddings of Graphs 3 Proof Every face has two possible orientations. You can order each face with an oriantation and neighboring faces get dierent ones. 2 Note, that the corollary 2 does not hold for non-orientable surfaces. Corollary 3 Answer to Question 2 is "yes" for Eulerian graphs. Corollary 4 Any 4-valent graph G is a medial graph of some orientable embedding. Proof Choose an Eulerian directed circuit in G and place the rotations around vertices so that the embedding is alternating. The alternating orientable embedding has bipartite dual. Choose faces of one color as vertices of the graph and join them by an edge if they share a vertex in G. 2 2 Some examples Figure 1: The pyramid graph and its medial graph - the antiprism A 4. For a start, let us give some examples of Eulerian embedded plane graphs and of plane graphs whose medial graph is Eulerian embedded. The most obvious examples of Eulerian embedded graphs are cycles Cn: The medial graphs of odd cycles, which are odd cycles with double edges, are also Eulerian embedded. There exist less trivial innite families of plane graphs, whose medial graphs are Eulerian embedded, too. It is easy to see, that the medial of the pyramid graph - the antiprism on Figure 1 is Eulerian embedded. We used the computer system Vega, see [3], to verify whether this property holds for all the pyramid graphs. We also checked the number of SAWs in medial

4 4 T. Pisanski, T. Tucker and A. Zitnik graphs of prisms P rn and antiprisms An. The results gave us the following theorem, which we state without proof: Theorem 5 s(an; Sphere) = s(m e(p r); Sphere) = ( 3 n = 3k 8 >< >: 1 n 6= 3k 1 n = 2k n = 4k 2 n = 4k + 2 s(m e(an); Sphere) = ( 4 n = 3k 2 n 6= 3k Let G 1 and G 2 be graphs, 2-cell embedded in orientable surfaces Sk1 and Sk2; respectively, where Sk denotes the sphere with k 0 handles. The connected sum G 1 #G 2 of graphs G 1 and G 2 with respect to the directed edges (v 1 ; u 1 ) in G 1 and (v 2 ; u 2 ) in G 2 is obtained as follows: take the union of graphs G 1 and G 2 and substitute the edges (v1; u1) and (v2; u2) by the edges (v1; v2) and (u1; u2): The rotation scheme is inherited from the embeddings of G 1 and G 2, except for the vertices v 1 ; v 2 ; u 1 and u 2 : In the rotation around v 1, u 1 is substituted by v 2, in the rotation around u 1, v 1 is substituted by u 2, and in the rotation around v 2, u 2 is substituted by v 1, in the rotation around u 2, v 2 is substituted by u 1. The connected sum of G 1 and G 2 is therefore a connected graph, 2-cell embedded in the surface Sk1+k2: The following theorem is very useful for constructing innite families of Eulerian embedded graphs: Theorem 6 Let G = G 1 #G 2. Then s(g; S) = s(g 1 ; Sk1)+s(G 2 ; Sk2)? 1: In particular, if G1 and G2 are Eulerian embedded, then G is Eulerian embedded as well. In Figure 2, the connected sum of the antiprisms A 4 and A 5 is shown. Both A4 and A5 are Eulerian embedded and so is their connected sum.

5 Eulerian Embeddings of Graphs 5 Figure 2: The connected sum of the antiprisms A4 and A5. Given an embedded graph, we substitute every k?valent vertex by a cycle on k vertices. The obtained graph is cubic and embedded in the same surface. It is called the truncation of the embedded graph. There are two types of faces in a truncated graph: the ones that correspond to former vertices and the ones that correspond to the faces with the boundary twice as long as in the original graph. In Figure 3, we show the number of SAWs of medial graphs of the graphs of platonic solids, their truncations and double-truncations We observe that the truncation of graphs sometimes preserves the number of SAWs in their medials. The graphs with this property are all cubic. It turns out, that this property holds for all cubic graphs. Graph Me Me(Tr) Me(Tr(Tr)) Tetrahedron Cube Octahedron Dodecahedron Icosahedron Figure 3: The numbers of SAWs in medial graphs of the platonic solids, their truncations and double-truncations. Theorem 7 The truncations of cubic maps preserve the number of SAWs in their medials. Proof Let us show, that for each straight-ahead walk in Me(G),

6 6 T. Pisanski, T. Tucker and A. Zitnik there exists a unique straight-ahead walk in Me(Tr(G)). To each vertex of G there corresponds a face in Me(G), which is in case of cubic graphs a triangle. In Figure 4 we can see a subgraph of Me(Tr(G)) that corresponds to a vertex of G, say v. It has a similar structure as a part of Me(G), corresponding to v, it only has another triangle inside. In each angle of the triangles, there are two edges going out. Let us denote the rst edge encountered when going from the boundary of the triangle in clockwise order, the rst edge, and the other edge, the second edge. To each edge going from the triangle in Me(G) there corresponds an edge going from the bigger triangle in Me(Tr(G)). Let us say, that the rst edges in Me(G) correspond to the second edges in Me(Tr(G)). Then a part of a straight ahead walk, going through the triangle in Me(G) has a unique correspondig straight-ahead walk in Me(Tr(G)). 2 Me?! # Tr Me?! Figure 4: What corresponds to a vertex of cubic graph in medial, truncation and medial of truncation So we obtain some other innite families of Eulerian embedded plane graphs - the medials of all the truncatios of the "odd" prisms, medials of their truncations and so on. 3 Eulerian medial embeddings A Petrie walk is a walk in an embedded graph G, where every two, but no three, edges lie on the boundary of the same face. It is not

7 Eulerian Embeddings of Graphs 7 hard to see that SAWs of medial graphs correspond to Petrie walks of the original map. See, for example, [2], where the Petrie walks are called left-right paths. If the signatures of edges in G are changed, ie. the orientation preserving edges become orientation reversing and vice versa, a dierent embedding of G is obtained, which is called the Petrie dual of (the embedded) graph G. The faces of the Petrie dual are exactly the Petrie walks of the original embedding of G. That means, that an Eulerian medial embedding of a graph is equivalent to Petrie dual being 1-face embedded. Theorem 8 Every graph embedding can be subdivided to give an Eulerian medial embedding. Proof The proof depends on the following idea: If SAWs of a 4-valent graph have two circuits at a vertex the other two matchings at a vertex give one circuit through that vertex. Subdividing an edge of the original graph can be viewed as changing the matching of the corresponding vertex of the medial graph. At each step we subdivide an edge, whose corresponding vertex of the medial graph is contained in two dierent SAWs, and at the end we obtain an Eulerian medial embedded graph. 2 The following corollary is an easy consequence of the Theorem and the fact that for every surface there exist medial graphs. Corollary 9 Every surface admits Eulerian embeddings. The question arises, whether every graph has an Eulerian medial embedding. If we consider only oriantable surfaces, the answer is "no". The simplest example of graphs having no orientable Eulerian embedding are even cycles. The embedding of an even cycle to an orientable surface is unique and the corresponding medial graph has two SAWs. But the answer is positive if we allow nonorientable embeddings, too. Theorem 10 For every rotation scheme, there is an assignment of signatures to edges that gives a medial Eulerian embedding (possibly nonorientable). Proof The proof is divided in two steps.

8 8 T. Pisanski, T. Tucker and A. Zitnik Change the signatures of edges between distinct faces until a oneface embedding is obtained. If the signature of an edge between two faces is changed, these two faces are merged to one face. The Petrie dual of the so-obtained graph has the medial with required property. 2 4 Eulerian embeddings of Eulerian graphs Every Eulerian directed graph has an Eulerian embedding, orientable and non orientable. To obtain such an embedding just choose any embedding where SAW is the given Eulerian circuit - at each vertex the opposite edges are consecutive in the Eulerian circuit. Let us state a theorem, characterizing Eulerian embeddings of covering graphs. For the denitions of covering graphs and Cayley graphs see, for example, [1]. Theorem 11 Let G be an Eulerian embedded graph, which is Eulerian directed according to its SAW. Given any cyclic voltage graph on G such that the sum of voltages along the directed edges generates the group, then the covering graph is Eulerian embedded. For the proof, see [4]. Cayley graphs are regular coverings of bouquets of circles. A regular embedding of a Cayley graph is given by lifting the rotation of the bouquet of circles to the Cayley graph. The rotation is called special, if the SAW in the bouquet of circles is Eulerian. The following theorem is thus an easy consequence of the Theorem 11. Theorem 12 Given any regular embedding of an even Cayley graph, it is Eulerian if and only if the group is cyclic, the rotation is special and the sum of the generators generates the group. 5 Conclusion and open problems Since each Eulerian graph G admits an orientable Eulerian embedding, it makes sense to dene an Eulerian genus for it. The natural question is which Eulerian graphs have their Eulerian genus equal to the

9 Eulerian Embeddings of Graphs 9 ordinary genus. Another question that can be posed is the following: which 2-cell embeddings of graphs have their connected and four-valent medial graphs Eulerian embedded? Finally, which graphs have at least one orientable embedding such that the corresponding medial graph is Eulerian embedded? Straight ahead-walks of medial graphs correspond to Petrie walks of the original map. Since maps that are Petrie duals to each other have the same 1-skeletons and the same medial graphs it would be interesting to explore the possibilities of extending this duality to straightahead walks of general embedded Eulerian graphs. References [1] Gross, J.L. & Tucker, T., Topological Graph Theory, J. Wiley & sons, [2] Lins, S., Richter, B. & Shank, H., The Gauss code problem o the plane, Aequationes Mathematicae, 33 (1987) [3] Pisanski, T. & coworkers, VEGA, a programming tool for manipulating discrete mathematical structures, see [4] Pisanski, T., Tucker, T.W. & Zitnik, A., in preparation.